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University  of  California  •  Berkeley 


The  Theodore  P.  Hill  Collection 
Early  American  Mathematics  Books 


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^^^l  94 


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.    WRITTEN  ARITHMETIC, 


COMMON    AND    HIGHER    SCHOOLS; 


TO    WHICH    IS    ADAPTED 


A   COMPLETE   SYSTEM  OF  REVIEWS, 


Ilf  THE   FORM   OF 


DICTATION  EXERCISES. 


BY 


G.   A.   WALTON, 

PRINCIPAL  OF  OLIVER  GRAMMAR  SCHOOL,  LAWRENCE,  MASS. 


BOSTON: 

1868. 


Entered,  according?  to  Act  of  Confrrcss,  in  the  year  1804,  by 

G.   A.  WALTON, 

In  the  Clerk's  Office  of  the  District  Court  of  tlie  District  of  Massachusetts. 


ELECTROTYPED  AT  THE 

BOSTON  STEREOTYPE  FOUNDRY, 

4  SPRING  LANE. 


PREFACE. 


This  book  is  designed  to  prepare  the  pupil  to  meet  the  demands  of 
actual  life.  It  is  itself  copious  in  examples  of  a  great  variety  of  forms, 
and  largely  of  a  practical  character  ;  and  the  accompanying  Key  con- 
tains a  set  of  Dictation  Exercises,  adapted  to  every  important  topic 
treated  in  the  book,  to  be  used  at  the  discretion  of  the  teacher,  by 
means  of  which  the  amoimt  of  practice  may  be  increased  almost  indef- 
initely. 

All  that  the  book  contains  is  written  for  the  pupil  j  and  if  he  will 
learn  it  understandingly,  he  may  master  the  principles  of  arithmetic 
with  but  little  aid  from  the  teacher. 

In  the  arrangement  of  subjects,  that  order  has  been  adopted  which 
experience  has  shown  to  be  the  best  for  all  classes  of  learners.  Some 
subjects,  of  little  importance,  have  been  briefly  treated ;  others  have 
been  transferred  to  the  Appendix.  Should  any  subject,  as  Duodeci- 
mals, Circulating  Decimals,  or  Average  of  Accounts,  or  any  examples 
prove  too  difiicult  for  the  younger  scholar,  they  can  be  omitted  till  the 
book  is  reviewed. 

Answers  are  given  to  the  examples,  so  far  as  is  necessary  to  assure 
the  pupil  that  he  understands  the  principles  ;  but  every  important  prin- 
ciple is  likewise  tested  by  examples  having  no  answers  in  the  book. 
The  answers  not  contained  in  the  book  may  be  found  in  the  Key, 
from  which  they  can  easily  be  transferred  to  the  black-board,  if  the 
teacher  prefers  to  have  them  placed  before  his  pupils. 

To  determine  the  adaptation  of  a  text-book  to  school  purposes,  it 

(3) 


4  PREFACE. 

must  be  used  in  the  school-room.  This  treatise  has  already  been  suc- 
cessfully tested  by  this  standard,  since  its  general  character  has  been 
determined  by  the  actual  demands  of  a  large  grammar  school,  at  pres- 
ent and  for  several  years  in  the  charge  of  the  undersigned,  and  since 
it  is  largely  illustrated  by  examples  which  have  been  repeatedly  em- 
ployed to  familiarize  students  with  the  principles  they  here  exempHfy. 
Its  practical  character  is  fully  certified  by  the  testimony  of  many  of  its 
students,  now  business  men,  who  practise  its  methods  in  the  office  and 
in  the  counting-room. 

Though,  at  the  request  of  the  publishers,  but  one  name  appears  upon 
the  title-page  as  author,  the  book  is  the  joint  production  of  the  person 
whose  name  it  bears,  and  of  E.  N.  L.  Walton,  former  teacher  in 
one  of  the  State  Normal  Schools  of  Massachusetts  ;  and  whatever  mer- 
its or  defects  the  book  may  be  found  to  possess,  may  be  attributed 
equally  to  each. 

Our  grateful  acknowledgments  are  due  to  many  teachers  and  business 
men  for  valuable  suggestions,  particularly  to  Wm.  J.  Rolfe,  A.M., 
of  Cambridge,  for  important  criticisms  while  the  work  was  in  prepara- 
tion for  the  press ;  to  Francis  Cogswell,  Esq.,  of  Cambridge,  for  hints 
on  methods  of  Reviews ;  and  to  the  teachers  of  the  Oliver  Grammar 
School,  Lawrence,  for  their  kind  assistance  in  solving  and  testing 

examples. 

GEO.  A.  WALTON. 

Lawrence,  Oct.  1, 1864. 


TABLE   OF   CONTENTS. 


SIMPLE  NUMBERS. 


Page 

Definitions, 9 

Notation  and  Numeration, 9 

Roman  Notation, 10 

Arabic  Notation, 12 

Numeration  Table,    . 14 

Fractional  Notation, 18 

Addition, 19 


Page 

Table  for  Practice, 22 

Subtraction, 25 

General  Review,  No.  1, 29 

Multiplication, 30 

Division, 3d 

Questions  for  Review, 41 

Miscellaneous  Examples, 42 


FEDERAL  MONEY. 

Table  for  Federal  Money, 45  i  Fundamental  Operations, 46 

Reduction, 46  |  Bills, 49 

Analysis, 53,  249, 254 

Questions  for  Review     54  |  General  Review,  No.  2, 65 

PROPERTIES  OF  NUMBERS. 


Definitions, 56 

Divisibility  of  Numbers, 57 

Table  of  Prime  Numbers, 59 


Factoring  Numbers,    .  .  , 
Greatest  Common  Divisor, 


61 


FRACTIONS. 


Definitions, 64 

General  Principles, 65 

Reduction  to  liOwest  Terms,   ....  6? 

Cancellation,    . 68 

Reduction  of  Whole  and  Mixed  Num- 
bers to  Improper  Fractions, ....  70 
Reduction  of  Improper  Fractions  to 
"Whole  or  Mixed  Numbers,  ....  70 

Multiplication  of  Fractions, 72 

Reduction  of  Compound  Fractions,  .  73 

Division  of  Fractions, 74 

Reduction  of  Complex  Fractions, .  .  77 


To  find  the  Whole  from  a  Part, ...  78 

What  Part  one  Number  is  of  another,  80 

Least  Common  Multiple, 81 

Common  Denominator, 84 

Addition  of  Fractions, 85 

Subtraction  of  Fractions, 86 

Greatest  Common  Divisor  of  Frac- 
tions,    88 

Least  Common  Multiple  of  Fractions,  88 

Questions  for  Review, 89 

Miscellaneous  Examples,  ......  91 

General  Review,  No.  3, 97 

(V) 


VI 


TABLE  OF  CONTENTS. 


COMPOUND  DENOMINA.TE  NUMBERS. 


Page 

Definitions, 98 

Federal  Money, 9S 

English  Money, 99 

Reduction, 100 

Comparison  of  English  and  Federal 

Money, 101 

Troy  Weight, 102 

Apothecaries'  Weight, 103 

Avoirdupois  Weight, 104 

Comparison  of  Weights, 105 

Long  Measure, 105 

Surveyor's  Measure, 107 

Mariner's  Measure, 107 

Cloth  Measure, 108 

Square  Measure, 108 

Cubic  Measure, Ill 

Liquid  Measure, 113 

Dry  Measure, 113 


Page 
Comparison  of  Liquid  and  Dry  Meas- 
ures,     lit 

Circular  Measure, 114 

Time  Measure, 116 

3Iiscellaneous  Table, 119 

Reduction  of  Fractions   to  Whole 

Numbers  of  Lower  Denominations,  120 
Reduction  of  Whole  Numbers  to  the 
Fraction  of  a  Higher  Denomina- 
tion,      121 

Compound  Addition, 123 

Compound  Subtraction, 126 

Table  of  Latitudes  and  Longitudes, .  130 

Compound  Multiplication, 132 

Compound  Division, 133 

Longitude  and  Time, 134 

Questions  for  Review, 135 

Miscellaneous  Examples, 138 


Duodecimals, 145  |  General  Review,  No.  4, 


149 


DECIMAL  FRACTIONS. 


Definitions, 150 

Numeration  Table, 150 

Addition, 153 

Subtraction, 154 

Multiplication  and  Division  by  10, 

100,  &c., 155 

Multiplication, 156 

Division, 157 

Reduction, 159 


Circulating  Decimals, 160 

Reduction  of  Compound  Numbers  to 
Decimals  of  a  Higher  Denomina- 
tion,      162 

Reduction   to   Whole   Numbers   of  • 

Lower  Denominations, 163 

Questions  for  Review, 164 

Miscellaneous  Examples, 165 


Peactice, 166  I  General  Review,  No.  5, 


PERCENTAGE. 


Definitions  and  Reductions, 171 

To  find  any  Per  Cent,  of  a  Number,  .  172 
To  find  100  Per  Cent,  from  a  given 

Per  Cent., 174 

To  find  what  Per  Cent,  one  Number 

is  of  another, 175 

Profit  and  Loss, 176 

Interest, 178 


Partial  Payments, 187 

Annual  Interest, 194 

Compound  Interest, 195 

Problems  in  Interest, 199 

Present  Worth  and  Discount, ....  202 

Banking, 201 

General  Review,  No.  6, .  208 

Commission,  Brokerage,  and  Stocks,  209 


TABLE  OF  CONTENTS. 


vu 


PERCENTAGE.  —  (Continued.) 


I»surance, 


Page 

.  213    Exchanjrc, 


Average  or  Equation  of  Payments,  .  215 

Average  of  Accounts 221 

Taxes, 225 

Custom  House  Business, 228 


Pago 
.230 


Questions  for  Review, 236 

Miscellaneous  Examples, 238 

General  Review,  No.  7, .  244 


MISCELLANEOUS. 


Ratio, 246 

Proportion, 248 

Analysis  and  Simple  Proportion,  .  .  249 
Analysis  and  Compound  Proportion,  254 

Simple  Partnership, 258 

Compound  Partnership, 200 

Questions  for  Review, 203 

Involution, 204 

Evolution, 205 

Square  Root, 206 

Application  of  Square  Root, 272 

Cube  Root, 276 


Mensuration, 283 

Circles,  Similar  Triangles,  Polygons,  293 

Similar  Solids, 295 

Questions  for  Review, 297 

General  Review,  No.  8, 299 

Alligation  3Iedial, 300 

Alligation  Alternate, 302 

Arithmetical  Progression, 305 

Geometrical  Progression, 309 

Annuities, 311 

Questions  for  Review, 314 

Miscellaneous  Examples, 315 


APPENDIX. 


Divisibility  by  9, 325 

Proof  of  Multiplication  and  Division 

by  casting  out  tlie  9's, 325 

Contractions  in  Multiplication,  .  .  .  320 

Contractions  in  Division, 828 

Exact  Interest  by  Days, 329 


Methods  of  Computing  Time, ...  .329 
Table  for  finding  Diiference  of  Days  331 
Mensuration  of  Timber,  &c.,    .  .   .  .332 

Gauging, 333 

Interest  at  7  3-10^, a35 

Metric  System, 336 


PUBLISHERS'  NOTICE. 


WALTON'S  ARITHMETICS. 

THE    SERIES    CONSISTS    OF    THREE    ROOKS,    VIZ.  : 

I.    Tlie    rictor-ial    Primary    -A^ritliinetlc. 
II.    Tlie    Intellectual    A.ritliinetlc. 
III.    Tlie    TV^ritten    .A.r'itliiiietic. 

The  publishers  invite  the  attention  of  Teachers  and  School  Officers 
to  this  series  of  Text-Books,  confident  that  on  examination  they  will 
commend  themselves  to  every  practical  educator.  No  other  series, 
in  general  use,  with  which  they  are  acquainted,  comprises  a  full 
course  of  Arithmetic  in  Three  Books. 

WALTON'S  DICTATION  EXERCISES 
are  supplementary  to  AValton's  Series,  and  afford  a  large  amount 
of  practice  in  the  fundamental  rules,  and  in  all  the  important  prac- 
tical applications  of  arithmetic.  They  are  designed  for  reviews  and 
test  exercises,  and  may  be  used  at  any  stage  of  the  pupil's  progress, 
and  in  connection  with  any  series  of  arithmetics. 


Course  of  Study. 


I.  Complete  the  Primary  Arithmetic  before  commencing  the 
Intellectual. 

II.  Complete  the  Intellectual  Arithmetic  to  page  63,  together 
with  written  exercises  [see  foot-notes  of  Intellectual],  before  com- 
mencing the  Written  Arithmetic. 

III.  Continue  the  study  of  the  Intellectual  in  connection  with  tlic 
Written  Arithmetic,  observing  to  complete  the  subjects  in  the  Intel- 
lectual before  commencing  them  in  the  Written. 


AEITHMETIC. 


Article  1,   Arithmetic  is  the   science   of  numbers,  and 
the  art  of  computing  by  them. 
S,    A  Unit  is  one. 
S.    A  Number  is  a  unit,  or  a  combination  of  units. 

4,  A  Concrete,  or  Denominate  Number,  is  a  number 
which  is  applied  to  some  object  or  objects ;  as,  one  hoy,  two 
apples f  three  slate  pencils,  four  sounds. 

5,  An  Abstract  Number  is  a  number  which  is  not  applied 
to  any  object ;  as,  one,  two,  three. 

6.   Exercise. 

Name  the  concrete  numbers  in  the  following  list :  — 

Four  girls ;  seven  swans ;  two ;  ten ;  nine  chairs  ;  five  knives  ;  eight ; 

twelve  horses  ;  six  mules ;  two  oxen ;  four ;  eleven ;  seven  pond  lilies ; 

one ;    ten ;   thirteen ;    nine    days ;    fifteen  lessons ;   two   rabbits ;   six 

bushels. 

Name  the  abstract  numbers  in  the  above. 

T.  The  fundamental  operations  of  written  Arithmetic  are 
based  upon  Notation,  and  consist  of  Addition,  Subtrac- 
tion, Multiplication,  and  Division. 

NOTATION  AND  NUMERATION. 

8.  Notation  is  the  art  of  writing  numbers. 
Numeration  is  the  art  of  reading  numbers. 
O.    Besides  being   expressed  in  words,  numbers  are  repre* 

(9) 


10  SIMPLE  NUMBERS. 

sented  by  letters  and  JigKres.  Tlic  method  of  representing  them 
by  letters  is  called  the  Roman  method,  because  it  was  used  by 
the  ancient  Romans.  The  method  of  representing  them  by 
figures  is  called  the  Arabic  method,  because  our  first  knowledge 
of  it  was  obtained  from  the  Arabs. 

Roman  Method. 

10,  The  Roman  Method  is  principally  used  in  writing 
dates,  and  in  numbering  chapters  and  sections  of  books. 

11,  It  employs  seven  capital  letters;  I  representing  one; 
Y,  five ;  X,  ten ;  L,  fifty ;  C,  one  hundred ;  D,  five  hundred ; 
M,  one  thousand. 

13,  By  combining  these  letters  in  various  ways,  all  num- 
bers may  be  expressed,  the  following  principles  being  ob- 
served :  — 

(1.)    When  a  letter  is  repented,  its  value  is  repeated, 

(2.)  When  a  letter  is  placed  before  another  of  greater  value, 
its  value  is  to  be  taken  from  that  of  the  greater;  thus,  IV 
denotes  four. 

(3.)  AVhen  a  letter  is  placed  after  another  of  greater  value, 
its  value  is  to  be  added  to  that  of  the  greater;  thus,  VI  de- 
notes six. 

(4)  AVhen  a  letter  is  placed  between  two  of  greater  value,  its 
value  is  to  be  tahen  from  their  united  value ;  thus,  XIX  denotes 
nineteen. 

(5.)  Any  letter  may  be  made  to  express  thousands  instead  of 
units  by  placing  a  dash  over  it.  Thus  X  denotes  ten  thousand ; 
B,  five  hundred  thousand ;  M,  one  thousand  thousand,  or  one 
million. 

13,    Table  of  Roman  Notation. 

denotes       five, 
six. 
seven, 
eight. 

♦  mi  is  sometimes  used  for  four. 


I       denotes 

one. 

V 

II 

two. 

VI 

ni 

three. 

VII 

IV* 

four. 

VIII 

NOTATION  AND  NUMERATION. 


11 


IX  *     denotes 

nine. 

L       denotes 

fifty. 

X 

ten. 

LX 

sixty. 

XI 

eleven. 

LXX 

seventy. 

XII 

twelve. 

LXXX 

eighty. 

XIII 

thirteen. 

XC 

ninety. 

XIV 

fourteen. 

C 

one  hundred. 

XV 

fifteen. 

CC 

two  hundred. 

XVI 

sixteen. 

CCC 

three  hundred. 

XVII 

seventeen. 

CD 

four  hundred. 

XVIII 

eighteen. 

D 

five  hundred. 

XIX 

nineteen. 

DC     . 

six  hundred. 

XX 

twenty. 

DCC 

seven  hundred. 

XXI 

twenty-one. 

DCCC 

eight  hundred. 

XXX 

thirty. 

CM 

nine  hundred. 

XXXI 

thirty-one. 

M 

one  thousand. 

XL 

forty. 

M 

one  million. 

XLI 

forty-one. 

MM 

two  million. 

14.   Exercises. 
Read  or  write  in  words  the  following  numbers:  — 


1.  IV. 

2.  XL 

3.  XIV. 

4.  XIX. 

5.  XXVL 

6.  XXIX. 

7.  XXXVL 

8.  XL. 

9.  XLV. 

10.  XLIX. 

11.  Lvm. 


12.  LXXL 

13.  LXXXIX. 

14.  XCVIII. 

15.  CLV. 

16.  CXIX. 

17.  CCCXLVIL 

18.  CDLXXIL 

19.  DCCXLIV. 

20.  MDXCIV. 

21.  MDCCCLXIV. 

22.  MD. 


15.   VTrite  the  following  in  Roman  characters :  — 

1.  All  the  numbers  from  one  to  twenty,  inclusive. 

2.  All  the  numbers  from  thirty  to  forty,  inclusive. 

3.  All  the  numbers  from  ninety  to  one  hundred,  inclusive. 

4.  One  hundred  thirty-eight. 


*  yill|  is  sometiines  used  for  nine. 


12  SIMPLE  NUMBERS. 

5.  Three  hundred  twenty-four. 

6.  Four  hundred  forty-nine. 

7.  Five  hundred  eighty-six. 

8.  Seven  hundi-ed  sixty-seven. 

9.  Nine  hundred  fifty-three. 

10.  One  thousand  four  hundred  seven.      _ 

11.  Five  thousand  eight  hundred.     Ans.  VDCCC. 

12.  Ten  thousand  ninety-nine. 

13.  One  thousand  eight  hundi-ed  sixty-four. 

Akabic  Method. 

16,  The  Arabic  Method  of  representing  numbers  employs 
ten  characters,  or  figures,  as  follows  :  — 

1,        2,        3,  4,        5,       6,        7,  8,         9,        0. 

One,  Two,  Thi-ee,  Four,  Five,  Six,  Seven,  Eight,  Nine,  Zero. 

17,  The  first  nine  are  called  digits,  from  the  Latin  word 
digitus,  ajlnger,  it  being  supposed  that  the  ancients  first  counted 
by  their  fingers.  They  are  also  called  signijicant  figures,  be- 
cause they  are  signs  for  numbers.  The  character,  0,  called 
zero,  signifies  nothing  when  it  stands  alone.  It  is  called  a  figure 
of  place  because,  in  writing  numbers,  it  is  used  to  fill  places  not 
occupied  by  other  figures. 

Used  singly,  these  characters  can  represent  only  the  numbers 
from  one  to  nine ;  but  combined  according  to  the  following  prin- 
ciples, they  are  used  to  represent  all  numbers. 

18,  The  figures  which  represent  simple  units  are  placed  at 
the  left  of  a  dot,  called  the  decimal  point.  (See  Art.  23  and 
note.)  The  first  place,  therefore,  at  the  left  of  the  decimal 
point  is  called  the  units'  place  ;  thus,  7.  is  read  "  seven  units,"  or 
"  seven." 

Having  no  single  figure  to  represent  ten  units,  we  consider  the 
collection  of  ten  units  as  one  ten,  or  a  unit  of  the  second  order^ 
and  represent  it  by  the  figure  1  put  in  the  tens'  place,  which  is 
the  second  place  from  the  decimal  point  towards  the  left ;  thus, 
10.  represents  ten,  the  zero  being  used  to  fill  the  units'  place, 
which  would  otherwise  be  vacant.     If  we  have  any  number  of 


NOTATION  AND  NUMERATION. 


13 


tens  and  units  to  write  together,  we  put  the  number  of  tens  in  the 
tens'  place,  and  of  units  in  the  units'  place;  thus,  thirty-six,  or 
three  tens  and  six  units,  is  written  36. 

A  collection  of  ten  tens  is  called  one  hundred,  or  a  unit  of  the 
third  order,  and  is  represented  by  1  in  the  third  or  hundreds'  place, 
(100.)  ;  two  hundreds  are  represented  by  2  in  the  hundreds'  place, 
(200.)  ;  three  hundreds  by  3  in  the  hundreds'  place,  (300.)  ;  etc. 

A  collection  of  ten  hundreds  is  called  one  thousand,  or,  a  unit 
of  the  fourth  order,  and  is  represented  by  1  in  the  fourth  or 
thousands'  place  (1000.)  ;  two  thousands  are  represented  '  y  2  in 
the  thousands'  place  (2000.) ;  etc. 


i  §  §  c  1 

H  a  H  p  Q 


7632. 
The  above  represents  seven  thousands,  six  hundreds,  three  tens, 
and  two  units,  and  is  read,  "  Seven  thousand  six  hundred  thirty^ 
two." 

Exercises. 
Read  the  followino: :  — 


19. 

1, 
2. 
3. 

4. 
5. 

20. 


86. 

132. 

6321. 

7862. 
99. 


7. 


10. 
Write  in  figures, — 

1.  One  thousand  six  hundred  forty-four. 

2.  Two  thousand  eight  hundred  twenty-one. 

3.  Nine  hundred  nine. 

4.  Six  thousand  two  hundred  ten. 

5.  Eight  thousand  eight. 

6.  Five  thousand  fifty. 

7.  Seven  thousand  seven  hundred  seventy. 

8.  Twenty-nine. 

9.  Six  hundred  two. 
10.  Six  thousand  twenty. 


428. 

11. 

9000. 

1302. 

12. 

9090. 

6006. 

13. 

9009. 

7801. 

14. 

8047. 

541. 

14  /  SIMPLE  NUMBERS. 

Questions.  —  What  is  the  first  place  at  the  left  of  the  decimal 
point  called  ?  What  is  the  second  place  at  the  left  called  ?  The  third  ? 
How  many  units  make  one  ten?  How  many  tens  make  one  hundred? 
How  many  hundreds  make  one  thousand?  How  many  units  make 
one  hundred?  How  many  units  make  one  thousand?  How  many 
tens  make  one  thousand?  What  are  units  of  the  first  order  called? 
Ans.  Simple  units.  What  are  units  of  the  second  order  called?  Of 
the  fourth  ?     Of  the  third  ? 

In  7632  how  many  tens,  and  what  number  remains  ?  How  many 
hundreds,  and  what  remains  ?     How  many  thousands  ? 

Hkmakk.  —  The  number  of  units  of  any  order  is  sometimes  called  a 
term ;  thus,  the  terms  of  632  are  6  hundreds,  3  tens,  and  2  units. 

SI,   Numeration  Table. 

CO  CO 

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*2  8  4,   9  6  ?,   3  4  r,   2  oT,   8  7  G,  3  2  2,  1  2   4 


TID  'CC  ~»o  "3.  fOjr  tSriS 

oo  oo  og  o*  o2  Oq 

'Si  'E  i  1:2  i  .2  -go  -C   rt 

f^S  P^^  e^j=  f^=  fSs  ;i:;s          p,c 

5-5  £«  £h  5M  S^  sI 


t3 


S^3*  The  ^i^/i  place  from  the  decimal  point  towards  the  left 
IS  the  ten  thousands^  place,  each  ten-thousand  being  equal  to  ten 
of  the  thousands  ;  iha' sixth  place  is  the  hundred  thousands'  place, 
each  hundred  thousand  being  equal  to  ten  ten-lhousands  ;  and  so 
on,  each'  unit  of  any  order  being  equal  to  ten  units  of  the  order 
immediately  preceding. 

We  now  see  that  the  number  of  units  of  any  order  is  expressed 
by  the  figure^  and  the  order  of  units  by  the  flace  which  the 
figure  occupies  ;  or,  in  other  words,  the  value  represented  by  any 


NOTATION  AND  NUMERATION.  15 

figure  depends  upon  the  figure  itself,  and  upon  the  place  which 
that  figure  occupies.  Thus,  2  in  the  first  place  means  simply 
two  (that  is,  two  units)  ;  in  the  second  place,  it  means  two  tens, 
or  twenty  ;  in  the  third  place,  two  hundreds. 

S3.  Since,  by  this  method  of  writing  numbers,  the  value 
represented  by  a  significant  figure  increases  as  that  figure  is  re- 
moved towards  the  left,  and  decreases  as  it  is.  removed  towards 
llie  right,  by  a  scale  of  tens,  the  system  is  called  the  Decimal 
System,  from  the  Latin  word  decern,  which  signifies  ten. 

Note.  — The  reason  for  calling  the  dot  (Art.  18)  a  decimal ijoint  must 
now  bo  obvious.  This  point  is  not  always  written,  but,  when  not  writ- 
ten, it  is  always  understood. 

24.  By  examining  the  table  (Art.  21),  we  find  it  sep- 
arated by  commas  into  groups  of  three  places  each.  These 
j]froups  are  called  periods,  the  first  period  being  that  of  units ; 
the  second  that  of  thousands ;  the  third,  millions ;  the  fourth, 
billions,  etc.  Thus  we  have  simple  units,,  tens  of  units,  and 
hundreds  of  units ;  units,  tens,  and  hundreds  of  thousands ; 
units,  tens,  and  hundreds  of  millions  ;  etc. 

52#>,    Exercises    ox  the  Table. 

1.  Give  the  names  of  the  first  two  periods  from  the  decimal  point, 
reading  them  towards  the  left ;  towards  the  right.  Give  the  names  of 
the  first  three  periods  in  the  same  way  ;  of  the  first  four ;  five ;  six  ; 
seven.  What  is  the  second  period  called  ?  third  ?  sixth  ^  seventh  ? 
fourth?  fifth? 

2.  In  which  period  are  found  thousands  ?  millions  ?  simple  units  ? 
trillions  ?  billions  ?  quintillions  ?  quadrillions  ? 

3.  In  which  place  of  what  period  are  found  tens  of  units  ?  thou- 
sands ?  hundred-thousands  ?  millions  ?  hundreds  of  units  ?  ten- 
thousands  ?  billions  ?  hundred-millions  ?  ten-billions  ?  ten-millions  ? 
quadrilUons  ?  ten-quintillions  ?  hundred-biUions  ?  hundred-trillions  ? 
quintillions  ?  ten-quadrillions  ?  ten-trillions  ?  hundred-quadrillions  ? 
trillions  ?  hundred-quintillions  ? 

4.  Name  the  order  of  units  of  each  number  in  paragraph  3.  Ans, 
Tens  are  of  the  second  order,  thousands  of  the  fourth  order ;  etc. 

5.  "What  order  of  units  is  found  in  the  first  place  of  the  second 
period  ?    Ans.  Fourth  order,  or  thousands.      In  the  third  place  of  the 


16 


SIMPLE    NUMBERS. 


first  period  ?  In  the  second  place  of  the  third  period  ?  In  the  third 
place  of  th  3  fourth  period  ?  In  the  first  j^lace  of  the  fifth  period  ?  In 
the  third  place  of  the  sixth  period?  In  the  second  place  of  the 
seventh  period  ?  In  the  third  place  of  the  third  period  ?  In  the  first 
place  of  the  seventh  period  ?  In  the  second  place  of  the  fourth  pe- 
riod ?     In  the  first  place  of  the  sixth  period  ? 

6.  In  6480921  how  many  tens,  and  what  remains  ?  Ans.  G48092 
tenSj  and  1  unit  remaining.  How  many  hundreds,  and  what  remains  ? 
Ans.  64809  hundreds,  and  21  remaining.  How  many  millions,  and 
what  remains  ?  thousands  ?  ten-thousands  ?  hundred-thousands  ? 

2G.  The  names  of  the  periods  employed  to  express  numbers 
higher  than  Quintillions  are,  in  their  order  from  Quintillions, 
Sextillions,  Septillions,  Octillions,  Nonillions,  Decillions,  Unde- 
cillions,  Duodecillions,  Tredecillions,  Quatuordecillions,  Quinde- 
cillions,  Sexdecillions,  Septendecillions,  Octodecillions,  Novende- 
cillions,  Vigintillions,  etc. 

37.    To  read  numbers,  observe  the  following 

Rule.  —  Beginning  at  the  umts'  place,,  point  off  the  expression 
into  periods  of  three  figures  each  ;  then  begin  at  the  left,,  and  read 
each  period  in  order  from  left  to  rights  giving  after  each,  excepting 
the  last,  the  name  of  the  period. 


38. 


Exercises. 

J.  Read 

or  write  in 

words  the 

follow  i 

ng:  — 

1. 

361. 

13. 

987654321. 

2. 

786. 

14. 

89743208. 

3. 

3261. 

15. 

1122334455. 

4. 

96321. 

16. 

3670980347. 

5. 

9301. 

17. 

9008007006. 

6. 

80021. 

18. 

12400496623. 

7. 

654237. 

19. 

245607000000. 

8. 

9326429. 

20. 

94632748632. 

9. 

9000200. 

21. 

1781006390800. 

10. 

86320029. 

22. 

62876432019623 

11. 

324867. 

23. 

753248734762869. 

12. 

81402020. 

24. 

943300896402798. 

See  Dictation  Exev'^ises,  Key. 


NOTATION   AND   NUMERATION.  17 

do.  Name  the  terms  in  the  first  example  above,  commencing  with 
units  (Art.  20,  Remark).  Ans.  One  unit,  six  tens,  three  hundreds. 
Name  the  tenns  in  tlie  second  example.  In  the  third.  In  the  other 
examples,  in  their  order. 

Head  from  the  Table  (Art.  21),  the  number  represented  by  the  first 
six  figures  from  the  decimal  point ;  the  first  eight ;  the  first  ten ;  nine ; 
twelve  J  fifteen;  seventeen;  twenty;  fourteen;  eighteen. 

S©,   To  write  numbers,  observe  the  following 

Rule,  —  Beginning  with  the  highest  period,  write  the  figures 

of  each  period  in  their  order  from  left  to  right,  filling  vacant 

places  with  :::eros, 

33.   Exercises. 

Write  the  following :  — 

1.  Three  hundred  sixty-four,  Ans,  364, 

2.  Seven  thousand  eighty-nine,  Ajis,  7089. 

3.  Eighteen  thousand  eighteen, 

4.  Nine  hundred  thousand  sixteen. 

5.  Four  hundred  twenty  thousand,  six  hundred  eighty- three. 

6.  Eight  hundred  ten  thousand,  two  hundred  four. 

7.  Two  hundred  fifty-nine  thousand,  seventy. 

8  Eorty-five  million,  seven  hundred  thousand,  two  hundred  fifty- 
one. 

9.  Nine  himdred  one  million,  two  hundred  eighteen  thousand, 
twenty-two. 

10.  Three  billion,  thirty-seven  million,  nine  hundred  six  thousand, 
two  hundred. 

11.  Two  hundred  thirty-four  million,  eight  hundred  sixty-three 
thousand,  three  hundred  eighty-nine. 

12.  Seventeen  billion,  seven  hundred  fifty-nine  million,  ninety  thou- 
sand, sixty-seven, 

13.  Three  hundred  thirty-three  quadrillion,  se\'en  hundred  seventy- 
nine  billion,  three  hundred  thousand,  two. 

14.  Nine  hundred  ten  quadrillion,  four  million,  three  thousand. 

15.  Fifty-four  quintillion,  eighty-three  quadrillion,  nine  hundred 
million,  seventeen  thousand,  one  hundred  eighty-two. 

16.  Eighteen  billion,  four. 

17.  Forty  million,  eight  hundred  thousand. 

2 


18  SIMPLE   NUMBERS. 

18.  Eighty-n'mc  million,  four  hundred  five  thousand,  seven. 

19.  Thirty-seven  trillion,  ninety-three  billion,  eighty-one. 

20.  Seven  hundred  quintillion,  one  quadrillion,  one. 

21.  Fifty  quintillion,  forty-nine  thousand,  thirty. 

3S*  We  have  seen  that  the  value  represented  by  a  figure  in- 
creases by  a  scale  of  tens,  as  the  figure  is  removed  towards  the 
left,  and  decreases  in  the  same  manner  as  it  is  removed  towards 
the  right.        ^ 

Applying  this  principle,  we  can  represent  parts  of  units  by 
placing  figures  at  the  right  of  the  decimal  point. 

If  we  consider  a  unit  to  be  composed  of  ten  equal  parts,  we 
may  represent  one  or  more  of  these  parts,  which  are  called  tenths, 
by  a  figure  in  the  first  place  at  the  right  of  the  point ;  again,  if 
we  consider  one  of  these  tenths  to  be  composed  of  ten  equal  parts, 
w^e  may  represent  one  or  more  of  these  parts,  which  are  called 
hundredths,  by  a  figure  in  the  second  place,  and  so  on. 

The  first  place  at  the  right  of  the  point  is  the  tenths'  place,  the 
second,  the  hundredths'  place,  the  third,  the  thousandths'  place. 

I     il 

^    IB    g   a 

E  £  -S  S 
*3   e   c  o 

Thus:  .7  8  5 
Here  the  7  at  the  right  of  the  point  represents  seven  tenths  of 
a  whole  one,  the  8  represents  eight  hundredths,  and  the  5  rep- 
resents five  thousandths.  The  entire  number  is  read  seven  hun- 
dred eighty-Jive  thousandths  ;  .25  is  read  twenty-jive  hundredths  ; 
.3  is  read  three  tenths. 

Exercises. 
Read  the  following :  — 

1.  .325;  .763;  .202;  .085;  .42;  .6. 

2.  .87 ;  .03 ;  .504  ;  .004 ;  39 ;  .039., 

Write  the  following :  — 

1.  One  hundred  three  thousandths.     Ans.  .103. 

2.  Eight  hundred  twenty-one  thousandths. 

3.  Two  hundred  forty-five  thousandths. 

4.  Seven  tenths.     Seven  hundredths. 


ADDITION.  19 


ADDITION. 

351.  Addition  is  the  process  of  finding  a  number  equal 
in  value  to  two  or  more  given  numbers  of  the  same  kind.  The 
number  thus  obtained  is  called  the  sum,  or  amount. 

An  upright  cross,  -f-  ,  read  plus,  is  the  sign  of  addition, 
and.  placed  between  t'/vo  numbers,  signifies  that  the  one  is  to  be 
added  to  the  other.  Two  horizontal  lines,  r=:  ,  read  equal  to, 
are  the  sign  of  equality,  and  signify  that  the  quantities  between 
which  they  are  placed,  are  equal ;  thus,  2  -[-  »^  =  7,  is  read,  two 
plus  Jive   is   equal  to  seven,  or,  two  plus  jive  equals  seven. 

Illustrative  Example. 
34.  Add  the  numbers  321,  285,  and  937. 

Opkkation.       We  first  write  these  numbers,  units  under  units, 
321         tens  under  tens,  hundreds  under  hundreds,  and  draw 
285         a  line  beneath.     Then,  adding  the  units  first,  7  -f-  5  -f 
937         1  =  13  units  =  1  ten  and  3  units  ;  we  write  the  3  in 
— ;; —         the   units'   place,   under   the  column  of  units,  and 
An:;.  1543         reserve  the  1  ten  to  add  with  the  column  of  tens. 
1  ten  +  3  tens  +  8  tens  +  2  tens  =  14  tens  =  1  hun- 
dred and  4  tens ;  we  write  the  4  tens  in  the  tens'  place,  and  reserve 
the  1  hundred  to  add  with  the  column  of  hundreds.     1  Imndred  +  9 
hundreds-}- 2  hundi'eds -[- 3  hundreds  =15  hundreds r=:l  thousanfl 
and  5  hundreds  ;  we  write  the  5  hundreds  in  the  hundreds'  place,  and 
the  1  thousand  in  the  thousands'  place,  and  thus  find  the  amount  of 
the   given   numbers   to    be  one  thousand  five   hundred  forty-three. 
Hence  we  derive  the  following 

Rule  for  Addition.  Write  the  numbers,  units  under  units, 
tens  under  tens,  hundreds  under  hundreds,  etc.  Begin  to  add  at 
the  units'  column.  If  the  sum  of  the  units  is  less  than  ten,  write 
it  under  the  column  of  units  ;  if  ten,  or  a  number  greater  than 
ten,  place  the  units^  f9^^^^  under  the  column  of  units,  and  reserve 
the  tens  to  odd  with  the  tens.  Proceed  in  the  same  way  with  the 
other  columns,  writing  down  the  entire  amount  of  the  last  column. 


20  SIMPLE  NUMBERS. 

Proof  I.  —  Add  each  column  in  a  reverse  direction ;  if  the 
same  result  he  obtained  as  before,  the  work  may  he  presumed  to  he 
correct. 

Note.  —  Greater  readiness  will  be  attained  by  mentioning  only  the 
results  in  adding  columns.  Thus,  in  the  above  example,  instead  of  saying 
7  and  5  are  12,  and  1  are  13,  say  7,  12,  13  ;  and  instead  of  saying,  1  ten 
and  3  tens  are  4  tens,  and  8  tens  are  12  tens,  and  2  tens  are  14  tens,  say 
1,  4,  12,  14  tens. 

35.     Examples  for  Practice. 

1.  What  is  the  sum  of  twenty-one,  sixty-seven,  eighty-nine, 
thirty-two,  forty-five,  thirteen,  ninety,  and  seventy-eight  ? 

Ans.  435. 

2.  What  is  the  sum  of  six  hundred  four,  nine  hundred  ninety- 
nine,  seven  hundred  ten,  six  thousand  nine  hundred  eighty-two, 
eleven  thousand  eight  hundred  seven  ?  Ans.  21,102. 

3.  What  is  the  sum  of  326,  981,  362,  707,  889,  and  864  ? 

Ans.  4129. 

4.  What  is  the  sum  of  246,  368,  909,  896,  763,  and  892  ? 

Ans.  4074. 

5.  What  is  the  sum  of  32689,  86543,  94861,  18325,  and 
90026?  Ans.  322,444. 

6.  What  is  the  sum  of  all  the  numbers  from  one  to  thirty, 
inclusive  ?  Arts.  465, 

7.  What  is  the  sum  of  all  the  numbers  from  one  hundred  fifty 
to  one  hundred  seventy-five,  inclusive  ? 

8.  Add  99,  364,  77,  86,  912,  32678,  96542,  and  32684. 

9.  Add  987,  5,  679,  369,  153,  888,  806,  17,  27,  and  5654. 

10.  Add  915,  875,  617,  868,  575,  387,  694,  946,  and  6377. 

11.  Find  the  sum  of  the  last  four  answers.  Ans.  189,506. 

12.  Add  987,  425,  672,  307,  216,  321,  111,872,564,876, 
318,  419,  187,  160,  and  3453. 

13.  875  +  466  -f  327  +  942  +  286  +  424  -f  309  +  429 
+  482  +  317  +  406  +  466  +  111  +  171  +  1618  =  what? 

14.  324  +  868  +  522  +  297  +  789  +  524  +  286  +  361 
+  472  +  884  +  472  +  287  +  649  +  592  +  1788  —  what? 

15.  876  +  205  +  918  +  468  +  207  +  948  +  572  +  618 
+  861  +  594  +  872  +  206  +  48  -f  500  +  918  + 1331  ==  what? 


ADDITION. 


21 


IG.  3G196  +  5384  +  2963  +  1200  +  100200  +  2560  + 
74  _|-  36  +  5  +  4786  +  186  +  544  +  396486  =  what? 

17.    Find  the  sura  of  the  last  five  answers.  Ans.  587,294. 

Proof  II.  —  Separate  the  example  into  two  or  more  parts  hy 
horizontal  lines  ;  add  the  parts  separately,  and  then  add  their 
amounts  ;  if  the  same  result  he  obtained  as  hefore^  ike  work  may 
he  presumed  to  he  correct.     See  Example  18. 


(18.) 

Proof. 

(19.) 

(20.) 

4163314 

7137500 

7984172 

5949841 

9345477 

8194324 

4956811 

1233198 

4221001 

1726414  16796380    2122172 

4754632 

9876431 

8914619 

3241320 

7325146 

3141691 

7987346 

9136' 

719 

4131261 

7325789 

8677485  35015781    3286432 

2941816 

^MS.  51812161  51812161    9710100 

2861423 

(21.) 

(22.) 

(23.) 

(24.) 

(25.) 

449 

9250 

81713 

247742 

3482 

788 

19 

93957 

303321 

6327 

435 

8158 

38 

478984 

8618 

663 

7901 

4885 

98517 

9532 

67 

6850 

3750 

232326 

2419 

455 

5102 

15 

879416 

4671 

399 

4372 

21901 

123192 

8384 

617 

3911 

86462 

10921 

3476 

31 

2514 

71557 

800467 

2123 

205 

1677 

99108 

93219 

619 

871 

3501 

8298 

63496 

9600 

431 

5528 

33984 

876201 

4520 

219 

7332 

67310 

23407 

5418 

868 

9415 

83568 

89467 

7317 

189 

8267 

97371 

77111 

2982 

598 

6408 

76503   ^ 

98121 

8415 

529 

4641 

86294 

267137 

3618 

721 

2286 

45939 

689642 

8976 

256 

3719 

36815 

232864 

6521 

583 

5931 

81541 

98518 

9357 

SIMPLE  NUMBERS. 


36  >   Table  for  Practice  in  the  Fundamental 
Operations. 


L 


21    20    19     18    ir    IG     15    14    13    12    11    10      9    8     7       (5     5     4       3     2      1 

A -9  8  7 

4  4  9 

0  2  5 

9  8  7 

9  15 

3  17 

1  8  4-A 

B-9  0  5 

7  8  8 

8  8  4 

4  2  5 

8  7  5 

9  9  5 

3  3  7-B 

C-6  7  9 

4  3  5 

5  6  8 

6  7  2 

6  1  7 

8  7  2 

6  9  2-  C 

D  -3  6  9 

6  6  3 

5  3  6 

3  0  7 

8  6  8 

3  2  7 

4  7  6-D 

E-1  5  3 

9  6  7 

8  5  6 

2  16 

5  7  5 

4  3  6 

2  0  8-E 

F-8  8  8 

4  5  5 

5  5  0 

3  2  1 

3  8  7 

2  16 

3  5  6-F 

G-8  0  6 

3  9  9 

5  16 

111 

6  9  4 

5  0  3 

5  7  5-G 

H-9  1  7 

5  1  7 

2  4  4 

8  7  2 

9  4  6 

8  4  2 

9  9  3-H 

1-927 

9  3  1 

3  3  0 

5  6  4 

6  4  ly 

'4  7  1 

8  8  8-^1 

J  -9  5  3 

2  0  5 

6  4  9 

8  7  6 

8  0  7 

17  4 

4  4  7-^  J 

K  -6  6  7 

8  7  1 

6  6  8 

3  18 

2  0  6 

8  6  2 

2  0  6-  K 

L  -7  2  8 

4  3  1 

6  6  9 

4  19 

5.2  6 

3  5  4 

4  1  7-  L 

M-8  4  5 

2  19 

2  4  8 

1  8  7 

8  4  9 

5  3  4 

6  2  1-M 

N-1  4  4 

8  6  8 

1  5  2 

16  0 

2  16 

111 

5  8  4-N 

0-225 

18  9 

3  9  3 

8  7  5 

45^ 

1  9  7 

2  9  0-  0 

P-1  9  9 

5  9  8 

9  8  2 

2  8  4 

5  7  5 

4  9  0 

9  4  0-P 

Q-9  4  1 

5  2  9 

5  5  5 

2  9  4 

18  0 

8  7  6 

7  0  0-Q 

R-7  9  5 

7  2  1 

8  2  3 

8  9  6 

5  9  4 

9  0  2 

2  9  8-11 

S  -7  3  4 

2  5  6 

3  2  1 

5  17 

8  0  6 

3  9  6 

4  7  5-  S 

T  -3  2  3 

5  8  3 

5  3  0 

8  9  4 

5  16 

4  8  4 

6  2  7-T 

U-8  2  4 

17  4 

7  7  4 

2  8  7 

4  9  9 

2  3  5 

5  8  6-U 

V-7  6  9 

4  16 

2  6  8 

4  0  6 

3  6  4 

3  8  7 

2  1  9-V 

W-1  0  8 

8  2  4 

8  2  4 

7  1  6 

7  6  2 

8  9  7 

9  8  4-W 

X~8  7  2 

8  9  2 

9  8  2 

8  7  2 

8  9  8 

6  2  4 

7  6  8-X 

Y-4  4  4 

7  6  4 

4  2  5 

5  7  4 

4  5  7 

4  7  6 

5  1  1-Y 

1* 


Add  in  the  above  Tiible'  (as  units,  tens,  and  hundreds) 

26.  1,  2,  and  3.       Ans,  13382.  I  "' ioi     13,  14,  and  1^  f 

27.  4,  5,  and  6.  \\1^''     ^l._16,  17,  and  18.^  ^/ 

28.  7,  8,  and  9.  I  \^^  ^ '  i^J\  19,  20,  and  21. 

29.  10,  11,  and  12.  J  j  C^?'0  33.''    1  to  21  inclusive. 
1^*  For  further  Exercises  on  the  Table,  see  Key 


'  H 

lU 
/'i^ 


ADDITION.  23 

34  Paid  $2400  (dollars)  for  my  farm,  $155  for  my  horse  and 
cart,  $26  for  garden  utensils,  $86  for  a  mowing-mathine,  $10  for 
a  horse-rake,  and  $108  for  a  pair  of  oxen.  Required  the 
amount. 

35.  A  body  of  troops  were  furnished  with  3622  Springfield 
rifled  muskets,  7690  smooth-bores,  and  13185  Enfield  rifles. 
Required  the  amount. 

36.  J.  R.  bought  of  the  Seneca  Knitting  Mills,  39600  pairs  of 
socks ;  of  Whitten,  Hopkins  &  Co.,  9782  pairs ;  of  Pierce 
Brothers  &:  Co.,  9353  pairs;  of  Allen,  Lane  &  Washburn,  5664 
pairs;  of  George  C.  Bosson,  4296  pairs;  of  Cushing,  Pierce  & 
Co.,  1315  pairs;  of  Samuel  Dennis,  276  pairs.  Required  the 
amount. 

37.  Required  the  average  number  of  pupils  attending  the 
Grammar  Schools  of  Boston  during  the  year  1859-60,  the  aver- 
age number  attending  the  Adams  School  being  493 ;  the  Bigelow 
School,  469;  Bowdoin  School,  538;  Boylston,  941;  Brimmer, 
575;  Chapman,  626;  Dvvight,  for  boys,  622;  Dwight,  for  girls, 
489;  Eliot,  708;  Franklin,  559;  Hancock,  719;  Lawrence, 
761;  Lincoln,  466;  Lyman,  370  ;  Mayhew,  367  ;  Phillips,  549  ; 
Quincy,  720  ;  Wells,  494  ;  Winthrop,  933. 

38.  In  the  year  1861,  Massachusetts  furnished  for  the  TJ.  S. 
army,  from  her  several  counties,  as  follows :  From  Barnstable,  3 
commissioned  officers  and  108  enlisted  men;  Berkshire,  21  offi- 
cers, 614  men;  Bristol,  59  officers,  1681  men;  Dukes,  0  officers, 
1  man;  Essex,  148  officers,  4134  men;  Franklin,  12  officers, 
482  men ;  Hampden,  35  officers,  845  men  ;  Hampshire,  15  officers, 
575  men;  Middlesex,  141  officers,  4200  men;  Nantucket,  1  offi- 
cer, 7  men;  Norfolk,  70  officers,  2031  men;  Plymouth,  44 
oificers,  1363  men;  Suffolk,  278  officers,  4111  men;  Worcester, 
110  officers,  3464  men.  Besides  these,  there  joined  her  regi- 
ments, 647  men  whose  residences  were  not  given,  and  20  oflficers 
and  955  men  from  other  States.  Required  the  whole  number 
of  enlisted  men  in  her  regiments ;  of  commissioned  officers ;  of 
both. 

39.  Massachusetts  furnished  army  shoes,  16649  -|-  4480  + 


24  SIMPLE  NUMBERS. 

7139  +  3228  +  2022  +  2336  +  2220  +  1000  +  1200  +  1236 
+  1013  +  240  pairs ;  cavalry  boots,  336  -f  1008  +  336  +  192 
-|-  160  -|-  168  -|-  150  pairs.  Required  the  number  of  pairs  of 
boots;  of  shoes;  of  both. 

40.  She  furnished  hats,  12000  +4704;  caps,  12130+  2934 
+  2069  +  450  +  251  +  98  +  160.  Required  the  number  of 
hats ;  of  caps. 

41.  On  commencing  business  a  merchant  had  $7752  in  cash, 
$7719  in  real  estate,  goods  valued  at  $9728,  a  lot  of  cattle  valued 
at  $6930,  a  ship  valued  at  $16834;  during  the  first  year  he  was 
in  trade  he  gained  above  all  his  expenses  $3195.  What  was  he 
worth  at  the  end  of  the  year  ? 

42.  What  is  the  number  of  square  miles  in  the  British  Isles, 
there  being  in  Scotland  30000,  in  England  31200,  in  Wales 
7200,  and  in  Ireland  32500? 

43.  The  United  States  contain  3284100  square  miles  more 
than  the  British  Isles  ;  required  the  area  of  the  United  States? 

44.  What  is  the  length  of  the  Grand  Trunk  Railway  from 
Detroit  to  Portland,  the  distance  from  Detroit  to  Stratford  being 
143  miles  ;  from  Stratford  to  Georgetown,  59  miles  ;  from 
Georgetown  to  Toronto,  30  miles ;  from  Toronto  to  Coburg,  69 
miles ;  from  Coburg  to  Belleville,  44  miles  ;  from  Belleville  to 
Kingston,  48  miles  ;  from  Kingston  to  Brockville,  47  miles  ; 
from  Brockville  to  Prescott,  12  miles ;  from  Prescott  to  Corn- 
wall, 46  miles ;  from  Cornwall  to  Montreal,  67  miles ;  from 
Montreal  to  Richmond,  73  miles  ;  from  Richmond  to  Island 
Pond,  71  miles  ;  from  Island  Pond  to  Gorhara,  58  miles  ;  from 
Gorham  to  Bethel,  21  miles  ;  from  Bethel  to  Danville,  42  miles  ; 
from  Danville  to  Portland,  28  miles  ? 

45.  How  far  is  it  from  Detroit  to  Toronto  ? 

46.  How  far  from  Toronto  to  Montreal  ? 

47.  How  far  from  Kingston  to  Montreal  ? 

48.  How  far  from  Montreal  to  Portland  ? 

49.  How  far  from  Portland  to  Gorham  ? 

50.  From  Boston  to  Portland  is  111  miles  ;  how  far  is  it  from 
Boston  to  Montreal  ? 

^P  For  Dictation  Exercises,  see  Key. 


SUBTRACTION.  25 


SUBTRACTION. 


37.  Subtraction  is  the  process  of  taking  one  number  from 
anotlier  of  the  same  kind,  to  find  the  difference. 

The  number  which  is  subtracted  is  called  the  subtrahend, 
from  the  Latin  suhtrahendus,  to  he  taken  from  under ^  as  that  is 
the  number  taken  away.  The  number  from  which  the  subtra- 
hend is  taken  is  called  the  minuend,  from  the  Latin  minuendus, 
to  he  made  smaller^  as  that  is  the  number  to  be  diminished.  The 
result  is  called  the  difference,  or  remainder. 

A  short  horizontal  line,  — ,  read  minus  or  less,  is  the  sign 
of  subtraction,  and,  placed  bet\veen  two  numbers,  signifies  that 
the  number  after  it  is  to  be  taken  from  that  before  it ;  thus, 
7  —  3  :rr  4,  read,  seven  minus  three  equals  four,  shows  that,  if 
3  be  taken  from  7,  the  remainder  is  4. 

Illustrative  Example,  I. 

38.  From  2G7  take  135. 

OPEaATioN.  Tor  convenience,  \vc  write  the  sub- 

Mmuend,       26/  trahend  under  the   minuend,  placing 

bubtrahend,  135  units    under   units,  tens  under   tens, 

T>        •  J         ion      4  hundreds  under  hundreds,  and  draw 

Bemamder,    132     Ans.  ,.      ,  i      ^       •      ,.         ^      • 

a  hne  beneath ;  5  units  from   /  units 

n:  2  units,  which  we  write  in  the  units'  place,  under  the  units  ; 
3  tens  from  6  tens  =  3  tens,  which  we  write  in  the  tens'  place ;  1 
hundred  from  2  hundreds  r=  1  hundred,  which  we  write  in  the  hun- 
dreds' place;  and  the  result  is  132,  which  is  the  difference  between 
267  and  135. 

39.  Proof.  If  132  is  the  difference  between  207  and  135, 
it  is  evident  that,  if  we  add  132  to  135,  the  sum  will  equal  267. 
Hence,  to  prove  subtraction,  add  the  difference  to  the  suhtrahend. 
If  the  sum  thus  ohtained  is  equal  to  the  minuend,  the  work  may  he 
presumed  to  he  correct. 

Note.  The  pupil  should  prove  each  example,  till  he  is  suro  that  he 
makes  no  nristakes. 


26  SIMPLE  NUMBERS. 


4:0.    Examples. 


4.  868879  —  42155  =  ? 

5.  974968  —  721265=:? 

6.  37879868  —  1244045:1:^? 


1.  368-^334  =  ?       ^«s.  34. 

2.  2769  — 2631  =  ?  ^775. 138. 

3.  362785-250122  =  ? 

Sum  of  the  last  four  answers  =  37828933. 

4L1.     Illustrative  Example,  II. 

9861— .3674  =  what? 

Oferation.  Here  a  difficulty  presents  itself.  We  cannot  take 
9861  4  units  from  1  unit.  In  order  to  i)erform  the  opera- 
3674       tion,  we  must  reduce  one  of  the  tens  in  the  minuend 

to   units,   which   with   the    1    unit   we   already   have, 

Ans.  G1S7  equals  11  units;  4  units  from  11  units  =  7  units, 
which  we  put  in  the  units'  place.  Having  reduced  one  of  the 
tens  to  units,  we  have  but  5  tens  left,  and  as  7  tens  cannot  be 
taken  from  5  tens,  we  must  reduce  one  of  the  hundreds  to  tens,  which 
r=  10  tens ;  10  tens  -j-  5  tens  =z  15  tens  ;  7  tens  from  15  tens  =  8 
tens,  which  we  write  in  the  tens'  place  ;  6  hundreds  from  7  hundreds 
n^l  hundred;  3  thousands  from  9  thousands :zr  6  thousands,  and  the 
answer  is  6187.     Hence  the 

lluLE  FOR  Subtraction.  Wiite  the  subtrahend  beneath  the 
mmueiidj  units  under  units,  tens  under  tens,  etc.  Begin  to  sub-, 
tract  at  the  units^  place,  taking  each  term  *  in  the  subtrahend  from 
the  one  above  it,  and  placing  the  remainder  beneath.  If  the  upper 
term  is  less  than  the  lower,  increase  it,  by  adding  to  it  one  of  the 
next  higher  denomination  reduced  to  its  oivn  denomination,  and 
then  subtract,  bearing  in  mind,  in  the  next  operation,  that  the 
upper  term  has  been  diminished  by  the  one  reduced. 

Examples. 
What  are  the  remainders  in  the  following  examples  ? 


(1-) 

(2.) 

(3.) 

(4.) 

Minuend, 

849 

321 

8642 

3084 

Subtrahend, 

278 

219 

730 

2427 

Remainder, 

571 

102 

7912 

657 

Proof, 

'  849 

321 

♦  See  Art. 

20.     Remark. 

SUBTRACTION.  27 

(5.)  (6.)  (7.)  (8.) 

From  3228  3256  7862  98731 

Take  409  2948  7589  19829 

Sum  of  the  last  four  remainders,  82,302. 

9.  A  man  had  375  oranges  in  a  box;  if  he  should  sell  259 
of  them,  how  many  would  he  have  left  ?  Ans.  116  oranges. 

10.  A  man,  having  451  acres  of  land,  gave  349  acres  to  his 
son  ;  what  remained  ?  Ans.  102  acres. 

11.  If  a  teacher  is  now  57  years  old,  and  has  taught  38  years, 
at  what  age  did  he  begin  to  teach  ?  Ans.  19  years. 

12.  How  old  was  a  person  in  1865  who  was  bom  in  1789  ? 

Ans.  76  years. 

13.  If  I  had  $625  in  a  bank,  and  withdrew  $249,  Avhat  re- 
mained? Ans.  $376 

4^.     Illustrative  Example,  III. 

From  20000  take  9. 
Opeuation. 
(I)  (9)  (9)  (9)  (10)        Here  we  have  no  tens  to  reduce  to  units,  no  hun- 
2  0  0   0  0     dreds,  and  no  thousands.   We  must  then  take  one  of  the 
9     2  ten-thousands  (leaving  1  ten-thousand),  and  reduce 
•  it  to  thousands,  making  10  thousands.     Reducing  one 

of  the  thousands  to  hundreds,  one  of  the  hundreds  to 
tens,  and  one  of  the  tens  to  units,  we  leave  9  thousands,  9  hundreds,  9 
lens,  and  have  10  units,  from  which,  if  we  take  9  units,  1  unit  will  re- 
main. Having  no  tens  to  take  from  9  tens,  no  hundreds  to  take  from 
9  hundreds,  no  thousands  to  take  from  9  thousands,  and  no  ten-thou- 
sands to  take  from  1  ten-thousand,  we  write  these  figures  in  their 
respective  places  below  the  line,  and  have  for  a  remainder  19991. 

43.     Examples. 

1.  From  2017  years  take  1028  years.  Ans,  989  years. 

2.  A  man,  who  had  1205  yards  of  cloth,  sold  429  yards.    How 
many  yards  were  left?  Ans.  776  yards. 

3.  There  are  205  sheep  in  a  flock  ;  if  109  of  them  should  be 
driven  to  market,  how  many  would  remain  ?  A?is.  96  sheep. 

4.  A  merchant  bought  goods  for  $1084,  and  sold  them  for  $177 
less  than  he  gave  ;  how  much  did  he  receive  for  them  ?  Ans.  $907. 


28  SIMPLE  NUMBERS. 

5.  30070  men  went  into  battle ;  4564  were  slain,  and  1300 
were  taken  prisoners ;  how  many  were  left  ?     Ans.  24,206  men. 

6.  Take  229  oxen  from  2006  oxen.  Ans.  1777  oxen. 

7.  Subtract  25  hundred  from  81  thousand.  A7is.  78,500. 

8.  How  many  more  in  47000  than  in  702  ?  Ans.  46,298. 

9.  47000  less  46298  equals  how  many  ?  Ans.  702.. 

10.  9832147  less  3472108  equals  how  many? 

11.  What  number  added  to  9213628  will  give  23475310  ? 

12.  What  number  subtracted  from  7654321  will  leave  369  ? 

13.  86293210  minus  329876  equals  how  many  ? 

14.  987621085  —  329875232  =z  how  many  ? 

15.  Find  the  sum  of  the  last  five  answers.   A7is.  771,984,860. 
i  6.  360080  +  7002  —  72824  =  what  ? 

17.  3478921  +  368754  —  2878796  =  what  ? 

18.  From  7654321  —  1234567  take  53899. 

19.  From4673214  +  2792  take  98264. 

20.  98432231  —  32636841  —  808994  =  what  ? 

21.  8087670  —  7549094  —  89699  =z  what  ? 

22.  Find  the  sum  of  the  last  six  answers.       Ans.  77,642,007. 

23.  What  is  the  difference  between  19360742  and  9643278? 

24.  How  many  times  can  I  take  7642  gallons  from  21002  gal- 
lons, and  what  will  remain  ? 

25.  If  the  minuend  is  36  quadrillion  and  the  subtrahend  95 
million  86,  what  is  the  remainder.     Ans.  35,999,999,904,999,914. 

26.  If  the  minuend  be  69  trillion  and  the  difference  85  bil- 
lion,   what  is  the  subtrahend? 

27.  Philadelphia  was  founded  in  1682.  In  what  year  was 
]S'ew  York  city  settled,  it  having  been  settled  68  years  before  ? 

28.  Victoria  ascended  the  throne  of  England  in  1837.  How 
many  years  has  she  reigned  ? 

29.  Napoleon  commenced  his  brilliant  career  in  1795.  How 
many  years  before  his  final  defeat  in  1815? 

30.  The  Israelites  left  Egypt  in  1491  B.  C.,and  40  years  after 
entered  the  land  of  Canaan.   In  what  year  did  that  event  happen  ? 

31.  In  the  year  1851,  London  had  2362000  inhabitants; 
Pekin  was  estimated  to  have  1500000.  How  many  more  inhab- 
itants had  London  than  Pekin  ? 


SUBTRACTION.  29 

32.  The  equatorial  diameter  of  the  earth  is  41843330  feet, 
and  the  polar  diameter  41704788  feet ;  required  the  difference. 

33.  The  population  of  St.  Louis  in  1850  was  77860,  and  in 
1860,   160773  ;  required  the  increase  in  10  years. 

34.  James  Nye  has  in  his  possession  $172  ;  he  owes  $28  to 
A,  $SQ  to  B,  and  $19  to  C.  After  paying  his  debts,  what  will 
remain  ? 

35.  I  have  saved  from  my  income  $362,  and  have  $2180  in 
government  bonds ;  how  much  more  must  I  save  that  I  may  pur- 
chase a  house  worth  $3500  ? 

4:4:,    General  Review,  No.  1. 

1.  Two  persons,  who  are  200  miles  apart,  travel  towards  each 
other,  one  46  miles,  the  other  51  miles  a  day ;  how  far  apart  will 
they  be  at  the  end  of  one  day  ? 

2.  If  the  above  persons  travel  away  from  each  other,  how  far 
apart  will  they  be  at  the  end  of  one  day  ? 

3.  A  man  gave  to  his  eldest  son  $3575,  to  his  youngest  son 
$4680,  and  to  his  daughter  $2495  less  than  to  the  youngest  son ; 
his  whole  property  was  worth  $20000 ;  what  sum  remained  ? 

4.  A  ship,  which  was  valued  at  $15590,  was  sold  at  a  loss  of 
$4975  ;  what  did  she  bring  ? 

5.  If  the  subtrahend  be  369  quadrillion,  and  the  remainder 
99  quadrillion    13  billion,   what  is  the  minuend  ? 

6.  The  difference  between  two  numbers  is  95478.  The 
larger  number  is  148769  ;  what  is  the  smaller? 

7.  How  many  times  can  18640  be  subtracted  from  46806, 
and  what  will  remain  ? 

8.  Which  of  the  two  numbers  15672  or  10560  is  nearer  to 
13465,  and. how  much  ? 

9.  From  what  number  must  846  be  taken  twice  to  leave 
15684? 

10.  To  what  number  must  962  be  added  three  times  to  make 
8472? 

11.  Which  is  nearer  to  348628,  63248  +  93264,  or  600063 
—  59321 ? 

O^  For  Dictation  Exercises,  see  Key. 


0Q  SIMPLE  NUMBEllS. 


MULTIPLICATION. 

4r5.  Multiplication  is  the  process  of  finding  a  number 
equal  in  value  to  one  number  taken  as  many  times  as  there  are 
units  in  another  number.  The  number  which  is  multiplied  is 
called  the  Multiplicand,  the  number  by  which  we  multiply  is 
called  the  Multiplier,  and  the  result  obtained  is  called  the 
Product. 

The  multiplicand  and  multiplier  are  often  called  factors  of 
the  product,  from  the  Latin  facio^  I  make,  because,  being  multi- 
plied together,  they  make  up  the  product.  The  product  is  also 
said  to  be  the  multiple  of  the  factors.  Thus,  7  tinaes  6  m  42, 
Here,  7  is  the  multiplier,  6  the  multiplicand,  and  42  the  product ; 
or  7  and  6  are  the  factors  of  42,  which  is  their  multiple. 

The  sign  of  multiplication  is  a  small,  oblique  cross,  X>  read, 
timeSf  or,  multiplied  hy.  Thus,  7X6  may  be  read  either 
7  times  6,  or  7  multiplied  hy  6.  In  the  former  case  7  is 
the  multiplier  and  6  the  multiplicand,  while  in  the  latter  6  is 
the  multiplier  and  7  the  multiplicand.  The  product  is  the  same, 
whichever  is  the  multiplier. 

Note.  —  In  the  process  of  multiplication,  the  multiplier  must  be  an 
nbstract  number.  We  cannot  multiply  pencils  by  pencils,  or  pencils  by 
apples,  but  either  may  be  multiplied  by  an  abstract  number,  and  give  a 
product  of  the  same  denomination  as  the  concrete  factor.     (Art.  4.) 

4:6.   Illustrative   Exampl*e,  I. 

Multiply  2364  by  7. 

Seven  times  4  units  i=:  28  units  r^ 
Operatiox. 

e    oQtiA  -MT  ^r  V      a  2  tens  and  8  units.     We  write  the  8 

„    ,       5    2364  Multiplicand,  ..,..,, 

p  actors  <  -  , ,  .y  y  in  the  units*  place,  and  reserve  th« 

(  7  Multiplier.  ^  «      ,  ,    , 

2  tens  for  the  tens  place.     7  times  6 

fHultiple,  16548  Product.  tens  =:  42  tens,  which,  with  the  2 

reserved  tens,  =  44  tensr=:4  hun- 
dreds and  4  tens ;  we  write  the  4  tens  in  the  tens'  place,  and  reserve 
the  4  hundreds  for  the  hundreds'  place.     7  times  3  hundreds  =zz  21 


MULTIPLICATION.  31 

hundreds,  which,  with  the  4  reserved  hundreds,  =:  25  hundreds  .zi:  2 
thousands  -\-  5  hundreds ;  we  write  the  5  hundreds  in  the  hundreds' 
place,  and  reserve  the  2  thousands  for  the  thousands'  place.  7  times 
2  thousands  m  14  thousands,  which,  with  the  2  thousands  reserved,  i= 
16  thousands  =  1  ten-thousand -j- 6  thousands;  we  write  the  6  thou- 
sands in  the  thousands'  place,  and  the  1  ten-thousand  in  the  ten-thou- 
sands' place,  and  thus  obtain  for  our  product  16548. 

Note.  —  This  result  might  be  obtained  by  finding  the  sum  of  the  num- 
ber 2364:  taken  seven  times;  that  is,  by  adding  2364  to  itself  six  times. 

Hence,  Multiplication  may  be  regarded  as  a  short  way  of  performing 
Addition. 

47.    Examples. 

Midtipli/ 

1.  267  by  2 ;  by  3  ;  by  4 ;  and  add  the  products.   Ans.  2,403. 

2.  628  by  5  ;  by  6  ;  by  7  ;   "         "  "  Ans,  11,304. 

3.  3401  by  8;  by  9;            "         "  "  Ans,  57,817. 

4.  90021  by  10;  by  11;     "         "  "  ^^^s.   1,890,441. 

5.  66285by  12;  by8;       "        "  "  Jbz*.  1,325,700. 

6.  4364  X  8  r=  what.? 


7.  7762  X  9  =  what? 

8.  5391  X  4  =  what? 

9.  3409  X  5  =  what? 


10.  9832  X    7  =  what? 

11.  8349  X     6  =  what? 

12.  22078  X  11  =  what? 

13.  19869  X  12  ==  what? 


14.  Add  the  last  eight  products,  and  multiply  by  7. 

Ans,  5,205,081. 

15.  123456  X  6  =  ?  16.     987654321  X  7  —  ? 
17.     Add  the  last  two  ;jroducts-  Ans,  6,914,320,983 

48.   Illustrative  Example,  II. 
Multiply  3648  by  294. 

Operatiox.  Here  we  are  to  multiply,  not  only  by  units,  but 

3648  ty  tens  and  hundreds.     We  write  the  numbers 

294  units  under  units,  tens  under  tens,  &c.,  and  mul- 

-lA^Q^  tiply  first  by  the  units,  as  before,  and  then  by  the 

32832  ^^^^      ^*  ^^  evident  that  the  product  of  any  num- 

7296  ^^^  multiplied  by  tens  will  be  ten  times  as  great 

as  if  multiplied  by  the   same  number   of  units ; 

1072512  Ans.  multiplied  by  hundreds,  one   hundred   times  as 
great  as  if  multiplied    by  units;   multiplied    hv 


32  SIMPLE  NUMBERS. 

thousands,  one  thousand  times  as  great,  etc.  Hence,  when  a  number 
is  multiplied  by  tens,  hundreds,  or  thousands,  the  products  thus  ob- 
tained are  written  one,  two,  or  three  places  farther  to  the  left  than  when 
multiplied  by  units  ;  or,  in  other  words,  we  multiply  by  the  other  terms 
as  we  multiply  by  the  units,  placing  the  fii'st  figure  of  each  product 
under  the  term  by  which  we  multiply.  The  sum  of  these  partial 
products  is  the  entire  product.     Hence  the 

Rule  for  Multiplication. 

Write  the  multiplier  under  the  multiplicand.  Beginning 
at  the  right,  multiply  each  term  of  the  multiplicand  hj  each  term 
of  the  multiplier,  successively,  placing  the  right  hand  Jigure  of 
each  partial  product  under  the  term  hy  which  you  multiply,  car- 
rying as  in  addition.  Add  all  the  partial  products,  and  the 
result  will  he  the  entire  product. 

4:^.  Proof  I.  Take  the  multiplicand  for  the  multiplier,  and 
the  multiplier  for  tlie  midtiplicand.  If  the  residt  thus  obtained  be 
like  the  first  result,  the  work  is  probably  correct, 

50.  Proof  II.  By  casting  out  the  9's.  Thia  method  is 
much  the  easier,  though  not  always  sure. 

Note.  —  To  cast  out  the  9's  from  any  number,  commence  at  the  hft^  and 
add  the  digits  towards  the  right.  When  their  sum  equals  9  or  more,  reject  9 
and  add  the  remainder  to  the  next  digit,  and  so  on.  The  last  remainder  is  called 
the  excess  of  9's. 

To  Prove  Multiplication  by  casting  out  the  9*s. 
Cast  out  the  d's  from  each  of  the  factors.  U^ien  multiply  the 
remainders,  shoidd  there  be  any,  cast  out  the  d^sfram  the  product,  and 
note  the  last  remainder.  Cast  out  the  d^sfrom  the  answer,  and  if  the 
remainder  equals  the  one  obtained  above,  the  work  may  be  presumed  to 
oe  right ;  thus, 

36184  3  +  6  =  0.    1  -f  8  =:  0.  4,  1st  remainder. 

2681     2  +  6  +  8  =  16z=04-7.     7-|-l  =  8,  2d  remainder. 

36184  32 

289472  3  +  2  zzz  5,  last  remainder. 

217104 
72368 


97009304  Ans.        7  -{-.  3  ir::  10  =  0  +  1.     1+4  =  5,  whicla  equal- 
ling  the  remainder  above,  the  work  is  right. 
Note.  —  For  demonstration  of  rule,  see  Appendix. 


MULTIPLICATION.  33 

Examples. 
51,     Perform  and  prove  the  following  examples:  — 


1.  3G84X  3G  =  ? 

Ans.  132,G24. 

2.  28-12  X  28  rir  ?    .^ 

Ans.  79,576. 


3.  18762    X  236=r? 

4.  128124  X  402=z:? 

5.  189003  X  836==? 

6.  12053    X  972  =  ? 


7.  Add  the  answers  to  the  last  four  examples,  and  multiply 
the  sum  by  3798.  Ans.  857,040,363,792. 

8.  Multiply  123456789  by  98765. 

«>2,  Any  number  may  be  multiplied  by  10, 100, 1000,  or  a 
unit  of  any  order,  hi/  annexing  as  many  zeros  to  the  multiplicand 
as  there  are  zeros  in  the  multiplier,  and  j)lacin(/  the  decimal  point 
at  the  right. 

Examples. 

9.  Multiply  68432  by  10,  by  100,  10000,  1000,  1000000, 
and  add  the  products.       ,  Ans.  69,192,279,520. 

10.  Multiply  3682  by  10000,  10,  1000,  100,  100000,  and  add 
the  products. 

03.    Illustrative  Example,  III. 
Multiply  68432  by  86000. 

OrERATIOX. 

68432  Here,  by  multiplying  first  by  86,  and  then  annex- 

86000  jj^g  three  zeros,  which  multiplies  the  first  product  b^ 

410592  one  thousand,  the  true  result  is  obtained,  and  labor 

547456  saved. 

5885152000  Ans. 

Illustrative    Example,  IV. 
Multiply  832000  by  210. 
Opkration. 
832000         Here  the  zeros  in  both  the  multiplicand  and  multi- 
210  plier  are  disregarded  until  after  muliplying  the  other 

gg2  terms  together. 

1664 

174720000  Ans. 
3 


34  SIMPLE   NUMBERS. 

54i,   Examples. 


11.  6320X80  =  ? 

12.  4682  X  360  =  ? 

13.  92873  X   86300=? 

14.  76000  X  8020  =  ? 

15.  32680  X  900100  =  ? 


16.  9876002  X  10001  =  ? 

17.  32001  X  20206  =  ? 

18.  987987  X  654653  =  ? 

19.  368043  X  77665=? 

20.  23698  X  84293  =  ? 


21.  Add  the  last  ten  answers,  and  multiply  the  sum  by  100. 

Ans.  81,482,871,584,800. 

22.  How  many  hills  of  corn  have  I  in  my  cornfield,  which  con- 
tains 97  rows  and  45  hills  in  a  row  ? 

23.  If  each  hill  produces  18  ears,  how  many  ears  does  the 
field  produce  ? 

24.  I  have  four  corn  bins,  containing  severally  63  bushels,  54 
bushels,  37  bushels,  and  29  bushels.  There  are  four  pecks  in  a 
bushel.     How  many  pecks  do  they  all  hold  ? 

25.  Allowing  23  ears  of  corn  to  a  peck,  how  many  ears  are 
there  in  the  bins  ? 

26.  If  a  barrel  of  flour  costs  9  dollars,  what  will  368  barrels 
cost? 

27.  If  a  person  by  working  12  hours  a  day  can  do  a  piece  of 
work  in  37  days,  in  how  many  days  can  he  do  it  working  1  hour 
a  day  ? 

28.  I  have  5  bins,  which  contain  69  bushels  each.  What  will 
be  the  capacity  of  a  bin  which  will  contain  as  much  as  all  of  them  ? 

29.  If  6  yards  of  cloth  will  make  one  pair  of  shirts,  how  many 
yards  will  make  one  dozen  or  12  shirts?  How  many  will  make 
8  dozen  ? 

30.  What  will  3  dozen  cost  at  15  cents  per  yard  for  the  cloth, 
30  cents  apiece  for  bosoms,  wristbands,  and  buttons,  and  50  cents 
apiece  for  making  ? 

31.  It  takes  7  yards  of  ticking  for  a  single  bed-sack;  what 
must  I  pay  for  cloth  for  18  single  bed-sacks,  at  16  cents  per 
yard  ? 

32.  If  sheeting  can  be  bought  for  17  cents  a  yard,  what  must 
I  pay  for  cloth  for  21  sheets,  allowing  10  yards  for  a  pair? 


DIVISION.  35 

33.  "What  will  be  the  cost  of  9  dressing  gowns  at  5  dollars 
apiece,  3  pairs  slippers  at  1  dollar  a  pair,  2  pairs  boots  at  4  dol- 
lars a  pair,  and  3  dozen  stockings  at  2  dollars  a  dozen  ? 

34.  Suppose  in  1  yard  of  cloth  there  are  580  fibres  of  warp 
^ind  432  of  filling,  and  that*  each  fibre  of  warp  contains  32 
strands,  and  each  of  filling  48,  how  many  strands  in  the  yard  ? 

35.  The  Lawrence  Pacific  Mills  turn  out  material  for  about 
65000  dresses  in  a  week ;  how  many  will  they  make  in  a  year^ 
or  52  weeks? 

36.  Allowing  12  yards  to  a  dress,  how  many  yards  do  they 
make  in  a  year  ? 

E^  For  Contractions  in  Multiplication,  see  Appendix. 
^^  For  Dictation  Exercises,  see  Key. 


DIVISION. 


S5m  Division  is  the  process  of  ascertaining  how  many  times 
^ne  number  is  contained  in  another,  or  of  finding  one  of  the  equal 
parts  of  a  number. 

Note.  —  In  the  example,  "  John  has  10  apples,  which  he  wishes  to  give 
to  as  many  boys  as  he  can,  giving  them  2  apples  apiece,  to  how  many  can 
he  give  them  ?  "  —  it  is  evident  he  can  give  them  to  as  many  boys  as  2  ts 
contained  times  in  10.  In  the  example,  ♦'!£  16  pears  are  divided  equally 
among  4  boys,  how  many  pears  does  1  boy  receive  ?  "  it  is  evident  that 
1  boy  must  receive  one  fourth  of  v/hat  the  4  boys  receive,  or  one  fourth  of 
16  pears  ;  that  is,  one  of  tliefour  equal  parts  of  the  number,  16  pears. 

The  number  which  is  divided  is  called  the  Dividend,  the  num  • 
ber  by  which  we  divide  is  called  the  Divisor,  and  the  result  the 
Quotient,  from  the  Latin  quoties,  how  many  times. 

The  sign  of  Division  is  a  short  horizontal  line  between  two 
dots,  -^  ;  thus,  9-^3  shows  that  9  is  to  be  divided  by  3.  Some- 
times the  dividend  and  divisor  take  the  place  of  the  dots;  thus,  f. 
This  expression  may  be  read,  9  divided  by  3,  nine  thirds,  or 
one  third  of  nine,  and  is  the  fractional  *  form  of  division. 

*  See  Art.  82. 


35  SIMPLE  NUMBERS. 

Short   Division. 

Note.  —  This   method   is   to   be   preferred  where   the   divisor  is   not 
greater  than  12. 

06.   Illustrative   Example,  I. 
Divide  936  by  6. 

Operation.  We  place  the  divisor  at  the  left  of  the 

Divisor  6)  93  G  Dividend,  dividend,  from  uhich  we  separate  it  by  .a 
— 7-  curved  line,  and,  drawing  a  straight  line 

Quotient     156  beneath  the  dividend,  proceed  thus  :   6  is 

contained  in  9  hundreds  1  hundred  times,  with  3  hundreds  remaining. 
We  write  the  1  hundred  beneath  the  hundreds  in  the  dividend,  and 
reduce  the  3  hundreds  remaining  to  tens.  3  hundreds  equal  30 
tens,  which,  with  the  3  tens  of  the  dividend,  equal  33  tens.  6  in  33 
tens,  5  tens  times,  with  a  remainder  of  3  tens ;  writing  the  3  tens 
in  the  tens'  place,  and  reducing  the  remainder  as  before,  we  have  36 
units.  6  in  36,  6  times  ;  writing  the  6  in  the  units'  place,  we  have 
156  as  the  quotient  of  936  divided  by  6. 

Illustrative   Example,  II. 
Divide  17869  by  7. 

Opekatiox.  In  this  example,  as  7  is  not  con  trained  in 

7  )  17869  1  (ten  thousand)  any  number  of  (ten  thou- 

.     '~Z77Z~  ^  T.^_  .,^^   sand)  times,  we  shall  have  no  ten  thousands 

Ans.   2oa2-5Ilemamder.  ,      /  i    ^u      p        .  -,       .^ 

m    the   quotient,    and    therefore   take    17 

(thousands)  for  our  first  partial  dividend.  We  find  also  that  the  div- 
idend does  not  contain  the  divisor  an  exact  number  of  times,  but  that 
there  is  a  remainder  of  5.  As  this  does  not  contain  7  any  whole 
number  of  times,  we  can  indicate  the  division  by  placing  the  5  in  the 
quotient  above  the  divisor,  and  have  for  the  answer  25524,  which 
is  read,  two  thousand  five  hundred  fifty-two  and  five  sevenths. 

From  the  above  examples  we  derive  the 

Rule  for  SfioiiT  Division.  Beginning  at  the  left,  divide  the 
•first  term  or  terms  of  the  dividend  hy  the  divisor,  make  the  residt 
the  first  term  of  the  quotient. 

Prefix  the  remainder,  should  there  he  any,  to  the  next  term  of 
the  dividend,  divide  as  before,  and  thus  cojitinue  till  all  the  terms 
of  the  dividend  are  divided. 

Should  there  be  a  remainder  after  the  last  division,  place  the 
divisor  beneath  it,  and  annex  the  remit  to  the  quotient. 


DIVISION.  37 

57,  Proof  I.  Division  is  the  converse  of  Multiplication, 
the  divisor  and  quotient  being  factors  of  the  dividend  :  hence,  to 
prove  an  example  in  division,  multiply  the  quotient  hy  the  divisor^ 
and  to  the  product  add  the  remainder.  The  sum  thus  obtained 
should  equal  the  dividend. 

«58.   Examples. 
Divide 


1.  3G945  by  3.  Ans.  12,315. 

2.  987654  by 4.  ^725.246,9131. 

3.  864024  by  6. 


4.  369801  by  9. 

5.  120087  by  11. 

6.  906102  by  3. 


7.  Find  the  sum  of  the  last  four  answers.      Ans.  498,044. 

8.  Divide  10101019  by  7. 

9.  Divide  16444006488  by  4. 

10.  23456983241  -^  9  =  ? 

11.  30089043921 -4- 7  n=  ? 


786491 
12. =  what? 

8 

369472 

13. =  what  ? 

o 


14.  How  many  barrels  of  flour,  at  7  dollars  a  barrel,  can  I 
buy  for  259  dollars  ? 

15.  At  11  cents  a  yard,  how  many  yards  of  cloth  can  I  buy 
for  368972  cents  ? 

16.  If  12  pieces  of  cloth  contain  408  yards,  how  many  yards 
in  a  piece  ? 

17.  Plow  many  weeks  are  there  in  4781  days  ? 

18.  How  many  hours  will  it  take  me  to  walk  lo78  miles,  at  5 
miles  an  hour  ? 

19.  9  times  a  certain  number  equals  324783 ;  what  is  that 
number?  Ans.  36,087. 

20.  8  X  what  =  36924?  21.  12  X  what  —  46817  ? 

Long   Division. 
•59,    Long  Division  is  the  process  of  dividing  where  the 
divisor  is  large,  and  the  work  written  down. 

60.   Illustrative   Example,  L 
Divide  85232  by  23. 


38  SIMPLE  NUMBERS. 

Opeuatiox.  23  is  contained  in  85  (thous.)  3  (thous.)  times ; 

23')85*'32(3705ii.    ^^  place  the  3  (thous.)  in  the  quotient,  at  the 

69  "  right   of  the    dividend.      3  (thous.)  X  23  =:  69 

~~  (thous.),  which,  subtracted  from  85  (thous.),  leaves 

a  remainder  of  16  (thous.).     Bringing  down  the 

next  figure  of  the  dividend,  we  have  162  (hund's), 

which  contains  23,  7  (hund's)  times  ;  we  place  the 
___  7  (hund's)  in  the  quotient  at  the  right  of  the  3 

17  (thous.).     7  (hund's)  X  23  =  161  (hund's),  which, 

subtracted  from  162  (hund's),  leaves  1  (hund.).  Bringing  down  the 
3  (tens)  of  the  dividend,  we  have  13  (tens),  which  does  not  contain  23 
any  number  of  (tens)  times.  Placing  a  zero,  therefore,  in  the  ten's 
place  in  the  quotient,  we  bring  down  the  next  figure,  2,  and  have  132 
units ;  23  in  132,  5  times.  "Writing  the  5  in  the  unit's  place  in  the 
quotient,  multiplying  and  subtracting  as  before,  we  have  a  remainder 
of  17,  and  for  our  answer,  3705^|.     Hence  the 

Rule  for  Long  Division.  Beginning  at  the  left,  divide  the 
first  term  or  terms  of  the  dividend  hy  the  divisor  ;  make  the  result 
the  first  term  of  the  quotient.  Multiply  the  divisor  hy  this  term, 
and  subtract  the  product  from  that  part  of  the  dividend  used. 

Annex  the  next  term  of  the  dividend  to  the  remainder  ;  divide 
as  before,  and  thus  continue  till  all  the  terms  of  the  dividend  are 
divided. 

Should  there  he  a  remainder  after  the  last  division,  place  the 
divisor  beneath  it,  and  cmnex  the  result  to  the  quotient. 

Note  1.  "WTien  it  is  difficult  to  determine  the  quotient  figure  at  sight, 
trial  divisors  may  be  used. 

For  example,  divide  29847  by  476. 

.„^  V     r.r..>.  ,  It  is  evident  that  476  is  contained  in  the  dividend 

476  ^  29847  (  • 

^  ^      fewer  times  than  400  is  contained  in  it,  and   more 

times  than  500.     Rejecting  two  right  hand  figures  from  the  divisor, 

also  from  that  part  of  the  divi4end  first  considered,  we  see  that  4  is 

contained  in  29,  7  times,  and  5  in  29,  5  times ;  therefore  the  quotient 

figure  cannot  be  more  than  7  nor  less  than  5. 

Note  2.  If,  at  any  time,  the  product  obtained  by  multiplying  the 
divisor  by  any  term  of  the  quotient,  exceeds  the  partial  dividend,  the 
quotient  figure  is  too  large.  If,  at  any  time,  the  remainder  equals  or 
exceeds  the  divisor,  the  quotient  figure  is  too  small. 


DIVISION.  Bd 

61.  Proof  II.  By  casting  out  the  9's.  (Art.  5[),  note.) 
Multiply  the  excess  of  9's  in  the  divisor,  by  the  excess  of  9's  in 
the  quotient,  and  find  the  excess  of  9's  in  the  product ;  if  it 
equals  the  excess  of  9's  in  the  dividend  after  the  remainder  has 
been  subtracted,  the  work  is  presumed  to  be  right. 

62.    Illustrative  Example,  II. 


Divide  26874  by  44. 

Operation. 

Proof. 

44)26874  (610  Quotient. 

4-f  4  =  8 

264 

6+  1  =  7 

47 

7  X  8  =  06.      5  +  6  =  11=^:0 

44 

+  2.     Casting  out  the  9,  we  not3 



the  remainder  2. 

34  Remainder. 

26874  —  34  zz:  26840. 

Placing  the  remainder,  34,  in 

2  +  6  +  8=16=0+7. 

a  fractional  form,  in  the  quo- 

7 +  4  =  11=0  +  2.    This  last  ro. 

tient,  we  have  for  the  answer. 

mainder,  2,  being  equal  to  the  first, 

610ff. 

the  work  is  right. 

63. 

Examples. 

1.  Divide  232848  by  56. 

Ans,  4158. 

2.  Divide  43572386  by  187.                            Arts.  233,007/gV. 

3.  Divide  18764321  by  262. 

4.  Divide  32456819  by  4618. 

5.  Divide  987654321  by  12345. 

6.  Divide  1459998  by  38,  19,  57,  171,  49,  513,  76842,  and 
add  the  quotients.  Ans.  182,075|f. 

7.  Divide  195989184  by  41,  16,  144,  164,  123,369,  72,  656, 
1968,  and  add  the  quotients.  Ans.  24,830,608. 

8.  Divide  -43586118576  by  17,  56,  119,  136,  4158,  51,  72, 
126,  99,  45738,  29106,  320166,  and  add  the  quotients. 

Ans.  6,288,215,934. 

9.  975318642  +-  893  =  ?      12.  8347300  +-  9004  =  ? 

10.  800000231  +-  73  =  ?    13.  769800281  -+  876  =  ? 

36845       ,  3987659002    ,  , 

11.  -— ^  =  what  ?  14.  — — -— —  =  what? 

268  83297 


40  SIMPLE  NUMBERS. 

04.     When  the  divisor  is  10,  100,  or  1000,  &c.,   we  can 

divide  by  simply  removing  the  decimal  point  in  the  dividend  as 

many  places  towards  the  left  as  there  are  zeros  in  the  divisor;  the 

number  at  the  right  of  the  point  will  he  the  remainder  ;    thus, 

368.4-  100  =  3.68  (or  ^-f^^).     Hence,  if  the  divisor  consists  of 

any  number  of  significant  figures  with  zeros  annexed,  first  cut 

off  the  zeros  from  the  divisor  and  an  equal  number  of  figures  from 

the  right  of  the  dividend,  then  divide  what  remains  of  the  dividend 

by  what  remains  of  the  divisor.      To  the  remainder,  if  any,  annex 

the  figures  that  were  cut  off  in  the  dividend  ;  thus, 

38|00)968|42(2o  ^.  y      ,.     ,  i      -,  .    ., 

'  -ft  Disregarding  the  tens  and  units,  we  find  now 

many  times   38  (hund.)  is  contained  in  968 

208  (hund.),  which  is  25,  with  a  remainder  of  18 

190  (hund.)  ;  this,  with  the  4  tens  and  2  units  left  in 

the  dividend,  makes  the  entire  remainder  1842, 

1842  ...  3800  is  contained  in  96842,  25^A1  times. 

60,     Examples. 

15.  Divide  42179  by  1000;  by  18000.  Ans.  42xWo  ;  ^fi^\% 

16.  Divide  76532102  by  48O0;  by  91000. 

17.  Divide  98000269  by  32600000;  by  980000. 

18.  Bought  a  farm  for  $18715,  at  ^95  an  acre;  how  many 
pcres  were  there  in  the  farm?  Ans.  197. 

19.  Paid  $4505  for  27  acres  of  woodland;  what  was  the  price 
per  acre  ? 

Solution.  If  27  acres  cost  $4505,  one  acre  will  cost  one  twenty- 
seventh  of  $4505,  which  equals,  &c. 

20.  Paid  $35328  for  368  acres  of  land;  find  the  price  per 
acre. 

21.  The  distance  from  Boston  to  Albany  is  198  miles,  from 
Albany  to  Buffalo,  298  miles.  How  long  will  it  take  a  train  to 
pass  over  the  road  at  the  rate  of  28  miles  an  hour,  allowing  2 
hours  for  detentions  between  Boston  and  Albany,  1  hour  at  Al- 
bany, and  3  between  Albany  and  Buffalo  ?  Ans.  23|^  hours. 

22.  The  Ohio  Canal  descends  1832  feet  in  152  locks ;  what  is 
the  average  descent  in  each  lock  ? 


DIVISION.  41 

23.  If  8  presses  can  coin  19000  pieces  of  money  in  an  hour, 
how  many  pieces  can  one  press  coin  in  a  minute,  60  minutes 
making  an  hour  ? 

24.  In  how  many  days,  of  12  hours  each,  can  the  president  of 
a  bank  sign  9000000  bank  notes,  if  he  signs  8  in  a  minute  ? 

^^^For  Contractions  in  Division,  see  Appendix. 
1^  For  Dictation  Exercises,  see  Key. 

I  66,     Questions  for  Review. 

1.  What  is  Arithmetic  ?  What  are  numbers  ?  What  is  an  ab- 
stract number  ?  a  concrete  ?  What  is  a  unit  ?  Define  Notation  and 
Numeration.  How  are  numbers  represented  ?  Describe  the  Roman 
method;  —  the  Arabic.  Which  is  more  used?  Why  is  this  sometimes 
called  the  Decimal  System  ?  What  is  the  decimal  point  ?  By  what 
is  the  number  of  units  of  any  order  expressed?  By  what  is  the  order 
of  units  expressed  ? 

2.  How  do  you  write  numbers  ?  How  do  you  read  numbers  ? 
Name  the  first  seven  periods.  Name  others  as  far  as  you  can.  How 
are  these  periods  separated  ?  What  are  the  names  of  the  places  of 
each  period  ? 

3.  What  is  the  least  number  of  figures  that  will  express  units?  — 
thousands  ?  —  billions  ?  —  trillions  ?  —  millions  ?  —  quadrillions. 

4.  In  189654238761,  what  is  the  largest  number  of  thousands?  — 
of  millions  ?  —  of  ten-millions  ?  —  of  hundred-billions  ?  —  of  trillions  ? 
—  of  tens  ?  —  of  hundreds  —  of  ten-thousands  ? 

5.  How  will  zeros  at  the  right  of  a  number  affect  it  ?  —  at  the  left  ? 

6.  What  does  a  figure  in  the  first  place  at  the  right  of  the  decimal 
point  represent?  —  in  the  second  place?  —  in  the  third? 

7.  What  is  Addition*?  What  is  the  sign  for  Addition  ?  —  for 
Equality  ?  How  do  you  arrange  numbers  to  be  added  ?  Is  this  ab- 
solutely necessary  ?  Perform  and  explain  an  example  containing  four 
numbers  of  at  least  seven  figures  each.     Give  the  rule. 

8.  AVhat  is  Subtraction?  Name  and  define  the  terms  used. 
What  is  the  sign  for  Subtraction?  Take  3684  from  7000068,  and 
explain.  Give  the  rule ;  —  the  proof.  When  the  minuend  and  differ- 
ence are  given,  how  can  you  find  the  subtrahend  ?  When  the  subtra- 
hend and  difference  are  given,  how  can  you  find  the  minuend? 

9.  What  is  Multiplication?  Name  and  define  the  terms  used. 
What  is  the  sign  for  Multiplication  ?  Perform  and  explain  an  example 
in  which  the  multiplier  has,  at  least,  two  figures.  Give  the  rule  •. — . 
first  method  of  proof  j  —  second  method.     How  do  you  multiply  by 


i2  SIMPLE  NUMBERS. 

10,  100,  1000,  &c.  ?  How  do  you  proceed  if  there  are  zeros  at  the 
Vight  of  the  multiplicand  or  multiplier?  Tens  X  units  z=:  what  ? 
Thousands  X  tens  ?  Hundreds  X  tens  ?  Ten-thousands  X  hundreds? 
Ten-thousands  X  ten-thousands? 

10.  What  is  Division  ?  Name  and  define  the  terms  used.  "What 
\s  the  sign  for  Division  ?  Perform  and  explain  an  example  in  short 
division.  Give  the  rule.  Give  the  proof  by  multiplication.  Perforn? 
and  explain  an  example  in  long  division.  Give  the  rule.  Give  the 
proof  by  casting  out  the  9's. 

11.  How  do  you  divide  by  10,  100,  1000,  &c.  ?  How  do  you  di- 
vide when  the  divisor  contains  zeros  at  the  right  of  significant  figures  ? 
When  the  dividend  and  quotient  are  given,  how  can  you  find  the 
divisor?  When  the  divisor  and  quotient  are  given,  how  can  you 
find  the  dividend  ?  When  the  multiplier  and  product  are  given,  how 
can  you  find  the  multiplicand.  When  the  multiplicand  and  product 
are  given,  how  can  you  find  the  multiplier  ? 

67.     Miscellaneous  Examples. 

1.  Add  nine  billion,  six  hundred  ninety-two  million,  eighty-one 
thousand  sixty-four ;  eighty-nine  trillion,  six  hundred  thirty-two 
million,  ninety-one  thousand  eighteen  ;  eighty-seven  thousand 
thirty-four  ;  and  two  hundred  sixty-eight  quadrillion,  nine  hun^ 
dred  eighty-four  trillion,  ninety-eight  million,  one  thousand 
ninety-four. 

2.  From  (900362840218  —  986234681)  take  (7682  + 
9619875.)  * 

3.  Multiply  (3684291  +  3642)  by   (8643264  —  8321628.) 

4.  Divide  (3687291  —  86)  by  (3684  +  232.) 

5.  If  892  is  one  factor,  and  28544  the  product,  \vhat  is  the 
other  factor  ? 

6.  365  times  what  number  z=  298570? 

7.  If  the  dividend  is  38493,  and  the  divisor  4277,  what  is  the 
quotient  ? 

8.  If  the  dividend  =  42777,  and  the  quotient  9,  what  is  the 
divisor  ? 

9.  There  were  52  schools  in  Antigua  in  1858,  with  4467 
scholars ;  required  the  average  number  in  each. 

*  In  examples  2,  3,  and  4,  first  perform  the  operations  indicated  within 
the  parentheses. 


MISCELLANEOUS.  43 

10.  David  was  born  1085  years  B.  C,  and  Washington  1732 
A.  D. ;  what  time  elapsed  between  these  events  ? 

11.  What  do  I  save  a  year,  my  income  being  $1600  a  year, 
and  my  expenses  $24  a  week,  52  weeks  making  the  year  ? 

12.  Illinois  produced  in  1860, 1515594  pounds  of  maple  sugar ; 
what  was  its  value  at  8  cents  per  pound  ? 

13.  Mississippi  produced  1195699  bales  of  cotton  ;  what  was 
its  value  at  13  cents  per  pound,  400  pounds  to  the  bale  ? 

14.  Missouri  produced  4164  tons  of  lead,  worth  $356660  > 
what  was  the  value  per  ton  ? 

15.  The  population  of  Chicago  in  1850  was  29963  ;  in  1860, 
109260  ;  what  was  the  average  increase  a  year  ? 

16.  If  8  men  can  do  a  piece  of  work  in  24  days,  how  long 
will  it  take  one  man  to  do  it  ? 

17.  If  768  be  one  factor,  and  861—237  the  other  factor,  what 
is  the  product  ? 

18.  Smith  &  Co.  consume  74  tons  of  coal  in  a  year ;  how 
much  more  must  they  pay  for  their  coal  in  1864,  when  coal  is 
$14  a  ton,  than  in  1860,  when  it  was  $8  a  ton? 

19.  From  the  invention  of  parchment  to  the  invention  of 
paper  was  782  years  ;  to  the  use  of  quills  in  writing  741  years 
more ;  to  the  invention  of  printing,  804  years  more ;  to  the  in- 
vention of  stereotyping,  345  years  more  ;  how  many  years  from 
the  invention  of  parchment  to  that  of  stereotyping  ? 

20.  Parchment  was  invented  887  years  B.  C. ;  when  was 
paper  invented  ?  Ans,  105  B.  C. 

21.  When  were  quills  first  used  in  writing  ?     Ans.  A.  D.  636. 

22.  When  was  printing  invented  ? 

23.  When  was  stereotyping  invented  ? 

24.  76854  divided  by  what  number,  gives  a  quotient  of  56 
and  a  remainder  of  22  ? 

25.  What  number  divided  by  87,  gives  a  quotient  of  3842  and 
a  remainder  of  76  ? 

26.  In  1853,  Wheeler  &  Wilson  made  799  sewing  machines; 
in  1854,  956;  in  1855,  1171;  in  1856,  2210;  in  1857,  45^1; 
in  1858,  7978  ;  in  1859,  21306  ;  in  1860,  19265  ;  in  1861, 
19725.     Required  the  amount. 


44  SIMPLE  NUMBERS. 

27.  If  a  sewing  machine  can  set  640  stitches  in  a  minute, 
how  many  can  it  set  in  an  hour?  —  in  a  day  of  12  hours?  — in 
6  working  days,  or  a  week  ?  —  in  52  weeks,  or  a  year  ? 

28t  There  Avas  sent  to  the  U.  S.  mint,  from  1823  to  1836, 
$4377984  worth  of  gold ;  what  was  the  average  value  sent  a 
year?  If  gold  was  worth  16  dollars  an  ounce,  how  many  pounds 
were  sent,  allowing  12  ounces  to  a  pound  ? 

29*  In  the  Pacific  Mills,  200000000  gallons  of  water  are  used 
in  a  day.  How  many  weeks  would  it  take  a  man  to  pump  it  if 
he  could  pump  a  gallon  in  six  strokes  of  the  pump,  20  strokes  a 
minute  for  16  hours  a  day,  allowing  6  working  days  per  week? 

30t  If  the  earth  is  95000000  of  miles  from  the  sun,  and  the 
moon  at  its  full  is  224000  miles  farther  on,  and  light  travels  at 
the  rate  of  191500  miles  a  second,  how  many  seconds  is  it  in 
passing  from  the  sun  to  the  moon  and  back  to  the  earth  ? 

Ans,  AdSj%\^^^%  seconds. 

31.  If  3871  be  divided  by  79,  and  the  quotient  be  multiplied 
by  133,  to  this  product  6523  be  added,  the  amount  divided  by 
40,  and  that  quotient  multiplied  by  970,  what  will  be  the 
product?  A?is.  316,220. 

32.    (17  — 2)^3=:?t  33.    (7  +  3)  X  2  =  ?t 

34.    (1803  4-7982)  X  3t_^       35.     19360 -^  9^+1  +  43t 

7  ""'  368  ~' 


36.*    (2  +  1  X  7  +  4)  ^  5  +  (8  +  6)  X  2  =  ?t 
37r    (81  +  9) -^10  +  67  +  (2  +  3X  7  +  7) -^6t=I? 
1^^  For  Dictation  Exercises,  see  Key. 

t  A  vinculum, ,  or  parenthesis  (^      ),  signifies  that  the  same  oper- 

Jition  is  to  be  performed  upon  all  the  quantities  thus  connected.  In 
solving  examples,  it  is  generally  better  first  to  reduce  all  quantities  con- 
nected by  a  vinculum,  or  parenthesis,  to  their  simplest  forms.  Thus,  in 
Ex.  32.  (17— 2) -f- 3  =15-^3=5.  Ex.  33. (7+3)  X  2  =  10  X  2  =  20. 
Ex.  36.  (2+  1  x7  +  4)~54-(8  4-6)X2  = 

(3  X  7  H-  4)  -i-  6  -i-;  14  X  2  =  5  -1-  28  =  33. 

Note.  — Examples  with  stars  are  "  optional  examples."  They  may  be 
omitted  by  younger  pupils  until  a  review,  or  altogether,  if  the  teacher 
prefers. 


FEDERAL  MONEY.  45 


FEDERAL    MONEY. 

68,  Federal  Money  is  the  medium  of  exchange  in  the 
United  States.  Federal  is  derived  from  the  Latin  fcedus, 
a  league ;  the  money  being  used  by  states  leagued  or  united 
under  one  government.  Federal  money  consists  of  eagles,  rep*, 
resented  by  E. ;  dollars,  represented  by  $ ;  dimes,  by  d. ;  cent&j. 
by  cts.,  and  mills  by  m. 

2hUe  of  United  States  Currency,  or  Federal  Money, 

10  m.  =    1  ct. 

10  cts.  r=    1  d. 

10  d.  =  $1 

$10  =    1  E. 

60,  As  these  denominate  numbers  increase  and  decrease 
like  simple  numbers,  hy  a  scale  of  tens,  they  are  written  as  sim- 
ple numbers  are  written,  and  operations  are  performed  upon  them 
as  upon  simple  numbers,  the  dollar  being  regarded  as  the  unit. 
The  sign  for  dollar,  $ ,  is  placed  before  any  number  which  we 
wish  to  designate  as  representing  United  States  currency. 

s  S  s  i  „• 

§=  =3  .§  g  2 

W  Q  «  o  S 

$89,445 
In  business  operations  the  denominations  eagles  and  dimes  are 
commonly  disregarded,  eagles  being  considered  tens  of  dollars, 
and  dimes,  tens  of  cents ;  thus,  the  above  illustration  is  read  8d 
dollars,  44  cents,  5  mills. 

Examples. 
70.     Write  the  following:  — 

1.  Seven    hundred    sixty-four  dollars  eighteen   cents  four 
mills.  Ans,  #764184. 

2.  972  dollars  17  cents  2  mills. 

3.  5768  dollars  9  cents  2  mills. 

4.  10  thousand  dollars  sixty  cents. 

5.  9  million  dollars  9  mills. 


Read  the  following:  — 

<^2789.842. 

11. 

$2009147.00. 

$9872.406. 

12. 

$98765481.052.- 

$9084.007. 

13. 

$4897.007. 

$864201.90. 

14. 

$987801.94. 

8329871.045. 

15. 

$81746.807. 

46  FEDERAL  MONEY. 

71 

6. 
7. 
8. 
9. 
10. 

16.  What  is  the  largest  number  of  cents  contained  in  exam- 
ple 6?  — 7?  — 8?  — 9?— 10?  IstAns.  278,984  cts. 

17.  What  is  the  largest  number  of  dimes  ? —  of  mills  ?  —  of 
eagles?  1*^  Ans.  27,898  dimes. 

18.  Read  examples  11  to  15,  making  cents  the  unit  of  numer- 
ation. 

19.  Reduce  $86452.  to  cents ;  to  mills. 

Ans.  8,645,200  cents  ;  86,452,000  mills. 

20.  Reduce  $9841.72  to  mills. 

21.  Reduce  8712647  cents  to  dollars. 

22.  Reduce  3687514  mills  to  dollars. 

How  do  you  reduce  dollars  to  cents  ?  to  mills  ? 
How  do  you  reduce  mills  to  dollars  ?  to  cents  ? 
How  do  you  reduce  cents  to  mills  ? 

23.  $9843.621  +  $4687.32  +  $84,321  +  $.07  -f-  $.64  -f 
$973,241  —  ?  A?is.  $15,589,213. 

Note. — In  addition  and  subtraction  of  Federal  Money,  dollars  should 
be  written  under. dollars,  cents  under  cents,  etc. 

24.  $3684.271  +  $765.42  +  $1763.417  +  $8645.217  — 
3.68  =  ?  .    Ans.  $14,854,645. 

25.  From    $8643.271   +    $98367.489    take    ($37,862    + 
$33695.41). 

26.  From  $3471.009  —  $.71  take  ($987,541  +  $862.73). 

27.  From  $4645.  +  $8178.  take  ($9827.  —  $6712.86). 

28.  $34865.002  X  46  =  ?  Ans.  $1,603,790,092. 

Note. — In  the  example  above,  as  mills  are  multiplied,  the  answer 
must  be  mills. 

29.  11  X  $3687.40  =  ?  31.   $98417.83  X  791  =  ? 
SO.    $946,918  X  478  zr:?               32.   984  X  $7654216.69=? 


FEDERAL  MONEY.  47 

Note  .  —  It  will  be  obvious  that  in  the  four  following  examples,  tht  quo- 
tient must  be  of  the  same  denomination  as  the  dividend. 

33.  $13428.  -^  9  =  ?  35.  $241364.48  -^-  56  =  ? 

34.  $7352.88  -^  12  =  ?  36.  $3712471.712  -i-ASS=z? 

^^^   Illustrative  Example. 
$1725. -M8=? 

Opekatiox. 
18)  1725  (  95.833^V  Ajis. 
162 

"TTt  In  this  example,  after  dividing  the  dollars,  we  have 

f.^  a  remainder  of  15  dollars;  this  we  reduce  to  dimes 

—  by  annexing  a  zero,  and   dividing,  obtain   8    dimes 

for  the  quotient  figure,  and  have  a  remainder  of  6 
}^^  .         dimes,  which  we  reduce  to  cents  and  divide,  and  have 
60         a  remainder  of  6  cents,  which  we  reduce  to  mills  and 
64         divide,  and  have  a  remainder  of  6  mills,  and  for  the 
(50       entire  quotient,  $95,833^%.  Ans. 
54 
6 
•Note.  — In  the  four  following  examples  continue  the  division  to  mills, 

37.  Divide  $9867.  by  37  ;  by  91  ;  by  416. 

38.  Divide  $89000.  by  17  ;  by  42  ;  by  368. 

39.  Divide  $36421.90  by  18  ;  by  48. 

40.  Divide  $6003489.  by  96  ;  by  543. 

41.  How  many  times  are  $.34  contained  in  $36.72? 

42.  How  many  times  are  $.25  contained  in  $645.  ? 

Note.  —  In  dividing  Federal  Money  by  Federal  Money,  when  the  de- 
nominations are  unlike,  it  is  necessary  first  to  reduce  the  dividend  and 
divisor  to  the  same  denomination.  The  answer  will  be  an  abstract  num- 
ber;  thus,  $645.  -J-  $.25  =r  64500  -^  25  =  2580. 

43.  Divide  $186432.18  by  $0,032. 

44.  Divide  $382971.21  by  $93. 

45.  Bought  1  pair  of  boots  for  $1.37;  1  pair  for  $1.65.; 
slippers  for  $.95  ;  shoes  for  $.65  ;  and  shoes  for  $.82.  Required 
the  entire  cost,  j^  r)\l\  ^| 

46.  Bought  a  horse  for  $95.00  ;  a  wagon  for  $63.00,  and 
harness  for  $  15.00  ;  kept  them  a  week,  paying  $  2.50  for  board 


48  FEDERAL  MONEY. 

for  the  horse,  then  sold  them  all  for  $  175.00.     Did  I  gain  or 
lose,  and  how  much  ?  ,S  \Q   c  Tj 

47.  What  cost  8  pairs  geese  at  $1.28  per  pair?  j^v|  i^^'^ 

48.  Bought  2  dozen  pigeons  at  $.85  per  dozen,  2  dozen  at 
$1.10  per  dozen,  and  1  dozen  for  $.90.     What  should  I  pay?  ^/7i 

49.  8874  sheep  were  sold  at  $4.13  per  head;  what  did  they 
bring  ? 

50.  There  were  shipped  to  Great  Britain  in  one  year  from 
New  York,  20602243  pounds  of  butter.  What  would  it  bring 
at  15  cents  per  pound  ? 

51.  39479897  pounds  of  cheese  were  shipped  the  same 
year.     Kequired  the  receipts  at  7  cents  per  pound  ? 

52.  4778  beeves  were  sold  in  New  York  market  in  one 
week,  averaging  874  lbs.  apiece,  at  7  cents  per  pound ;  what 
was  received  for  them  ? 

53.  Bought  2  pieces  of  flannel,  each  containing  62  yards,  for 
$39.68,  and  sold  them  for  40  cents  per  yard.    What  did  I  gain  ? 

54.  Paid  a  man  $16.25  for  13  days'  work;  what  was  that 
a  day? 

bb.  Paid  $5.10  for  17  boxes  strawberries;  what  was  that 
a  box? 

56.  Among  how  many  boys  may  $10  be  distributed,  that 
each  may  receive  $0,625  ? 

57.  Sold  35  barrels  Greenings  at  $1.75  per  barrel,  17  barrels 
Baldwins  at  $1.80  per  barrel,  12  barrels  fall  Harveys  at  $1.25 
per  barrel,  and  25  of  Russets  at  $2.25  per  barrel.  Paid  17  cents 
a  barrel  for  picking,  and  $12.00  for  transportation.  What  re- 
mained after  all  my  bills  were  paid  ? 

58.  Paid  $3.00  for  1  dozen  apple  trees,  $3.36  for  1  dozen 
peach  trees,  $3.30  for  one  half  dozen  pear  trees ;  what  did  I 
pay  for  the  whole,  and  how  much  a  piece  for  each  kind  ? 

59.  Paid  a  carpenter  for  stock  and  work  for  a  house,  $450.75  ; 
for  mason's  work,  $38.25  ;  for  digging  and  stoning  cellar,  $47.18 ; 
for  painting,  $40.00  ;  to  the  plumber,  $8,125.  I  then  sold  it, 
and  lost,  in  so  doing,  $14,305  ;  what  did  I  sell  it  for  ?   Ans.  $570. 


FEDERAL  MONEY.  49 

60.  Bought  a  farm,  containing  40  acres  meadow  and  17  wood- 
land, for  $2850.00.  Sold  to  one  man  10  acres  woodland  for 
$85.00  per  acre ;  to  another  a  house  lot  of  one  acre  for  $90.00  ; 
and  the  remainder  to  a  third  for  $2025.00.  What  did  I  gain 
by  the  operation;  and  for  how  much  per  acre  did  I  sell  the  re- 
mainder? Ans.  $11^  i  $44.02^85. 

Bills. 

73.  When,  in  a  business  transaction,  one  person  receives 
money,  property,  or  services  from  another,  he  becomes  indebted 
or  is  debtor  for  the  amount  he  receives. 

The  person  who  parts  with  the  money,  property,  or  services, 
is  credited  for  the  amount  he  has  given,  and  hence  is  called  the 
creditor. 

A  written  statement  of  the  amount  of  the  debt,  with  the  items 
included,  is  called  a  hill,  and  is  usually  written  in  forms  like 
those  on  the  following  pages. 

When  the  creditor  is  paid  the  amount  due,  he  acknowledges 
the  receipt  by  his  signature  at  the  foot  of  the  bill,  after  the 
words  "  Received  payment."  A  bill  thus  signed  is  said  to  be 
receipted, 

74 •  Find  the  cost  of  each  article  in  the  following  bills,  and 
their  several  amounts. 


Mr.  James  Crocker 
10  bbls.  St.  Louis  Flour, 

Buffalo,  November  10,  1862. 

> 

Bought  of  Henry  Shedd, 

extra,               at       $  9.50 

12     "      Western       « 

medium,           " 

7.75 

8     "      Canada        " 

extra,                " 

6.72 

14     «      Canada        « 

choice  extra,    *' 

7.87 

3     «      Corn  Meal, 

it 

4.25 

20   bu.    Northern  Oats, 

a 

.61 

Received  payment, 


$376.89. 


Henry  Shedd, 

By  George  Ba^^ 


50 


FEDERAL  MONEY. 


(2.) 

Lawrence,  November  18,  1862. 

Mr.  D.  Danforth, 
65  bu.  Potatoes,          at 

Bought  of  J.  Smith, 
$0.55 

300  lbs.  Squashes,          " 
450    «    Pork,                 « 

.01 
.11 

35  bu.  Beans,               " 

2.50 

85    «    Rye  flour,         " 

2.25 

Received  payment, 

$367.00  Ans. 

J.  Smith. 

(3.) 

Boston,  April  17,  1863. 

Mr.  James  Blake, 

Bought  of  Breck  &  Co., 

3  bu.  Herds  Grass,      at 

$2.25 

75  lbs.  Clover  Seed,        " 

.11 

25  bu.  Canary  Seed,      " 
18  lbs.  Mustard  Seed,     « 

3.62 
.13 

25   «     Hops,                   « 
22   "     Hops,                   " 

.17 
.16 

Received   payment. 

$115.61  Ans. 

Joseph  Breck,  for  Breck  &  Co. 

(4.)  New  Bedford,  October  9,  1862. 

Mr.  J.  L.  Rice, 

To  Henry' Brown,  JDr.* 

To      2  bbls.  Pork,  prime,   at    $  15.50 

«   250    lbs.   Hams,  «  .09 

«   475     "     Butter,  «  .24 

«   4^2     "         «  «  .18 

Received   payment,  $ 

Henry  Brown. 

means  that  Mr.  Rice  is  debtor  to  Mr.  Brown.    Dr,  is  rea4 


BILLS.  51 


(5.) 

New  York, 

,  March  7,  1868. 

A.  M. 

Phipps,  Esq., 

To  Samuel  Sloane,  Dr, 

Jan.    7. 

To 

12  lbs.  Tartaric  Acid,       at 

$   .85 

ii      ii 

ii 

7     "    Blue  Vitriol,          " 

.25 

«     12. 

a 

3    oz.  Morphine,              " 

7.00 

"     13. 

ii 

5     "    Quinine,                 " 

4.00 

Feb.   2. 

ii 

2  lbs.  Cardamoms,           " 

3.50 

a         a 

it 

10     «    Cream  Tartar,       " 

.51 

ii          a 

a 

8     «    Cubebs, 

.53 

a         a 

a 

5     "    Gum  Copal,           " 

.68 

ii         ii 

a 

8galls.Cod  Liver  Oil,      " 

1.75 

Received   payment,  $ 

Samuel  Sloane. 


(6.)* 

Bristol, 

January  1, 

1863. 

Otis  Butler,  Esq., 

1862. 

To  Ralph  Burnside, 

Dr. 

Apr.  3. 

To    96  lbs.  Rice, 

at 

$  .07 

a       a 

"        3   «    Saleratus, 

li 

.08 

a       a 

"      28   «    Castile  Soap, 

ii 

.16 

May  9. 

"      25   «    Pearl  Starch, 

a 

.09 

«    15. 

"    196   «    Crushed  Sugar, 

a 

.13 

a      a 

«    196   «    Brown  Havana, 

ii 

.12 

June  5. 

«      46   "    Hyson  Tea, 

a 

1.12 

July  10. 

"     37   "    Gunpowder  Tea, 

a 

.95 

$ 

Cr.t 

May  10. 

By  1  Wagon,                      $42.00 

"     16: 

"    2  Cows,  at  $35.00, 

ti        a 

"       Cash,                             10.00 

$ 

$. 

Received  payment, 

•♦r  • 

Ralph  Burnside. 

t  This  means  that  Mr.  Butler  is  credited  for  goods  or  cash  delivered. 
Cr.  is  read  "  creditor." 


52  FEDERAL  MONEY. 

75.  Find  the  amounts  due  in  the  following  examples,  and 
make  out  the  bills. 

7.  Charles  Fuller  purchased  of  James  Munroe,  Jan.  4,  1863, 

1  horse  for  $95.00,  2  cows  at  $50  apiece,  1  wagon  for  $62.00, 

2  shovels  at  $1.12  apiece,  30  bushels  corn  at  $.65  per  bushel, 
and  17  bushels  wheat  at  $1.62  per  bushel. 

8.  Samuel  Banks  sold  to  Abraham  Seward,  March  10,  1863, 
2  pieces  flannel,  of  62  yards  each,  at  $.49  per  yard ;  5  pieces 
cotton,  bleached,  at  $.24  per  yard,  2  of  the  pieces  containing 
36  yards  each,  and  3  containing  35  yards  each;  38  yards  ticking, 
at  $.29  ;  86  yards  brown  sheeting,  at  $.27 ;  42  yards  broadcloth, 
at  $3.65  per  yard. 

9.  Dr.  Cardamom  bought  of  James  Mortar  3  gallons  castor-oil  at 
$2.50  ;  9  pounds  oil  peppermint  at  $2.50  ;  4  pounds  oil  cassia  at 
$3.62  ;  4  pounds  oil  orange  at  $3  ;  6  pounds  oil  lemon  at  $4.25 ; 
5  pounds  oxalic  acid  at  $.33  ;  and  5  pounds  Seneca  root  at  $.95- 

10.  Baldwin  &  Lewis,  of  Cincinnati,  bought  of  Balch  &  Ray- 
ner,  Boston,  24  sack  coats  at  $15.75  ;  36  vests  at  $3.50 ;  95 
pairs  pants  at  $4.38 ;  4  dozen  pairs  suspenders  at  42  cts.  per  pair ; 
23  dozen  pairs  gloves  at  68  cts.  per  pair;  and  15  dozen  collars 
at  13  cts.  apiece. 

11.  Hiram  Teachwell  bought  of  Mark  Thrifty,  Nov.  8,  1862, 
2  Dictionaries,  at  $.87  apiece;  9  Yocal  Cultures,  at  $.70;  12 
Walton's  First  Steps,  at  $.13,  and  24  Worcester's  Spellers,  at 
$.20.  Dec.  2,  he  bought  2  reams  paper  at  $2.12,  3  dozen  pen- 
cils at  $.50,  and  12  slates  at  $.17.  Dec.  10,  he  paid  Mr.  Thrifty 
$20.00,  and  Jan.  1,  1863,  Mr.  Thrifty  made  .out  his  bill.  Re- 
quired  the  balance  due. 

12f  Solomon  Katchall  bought  of  Hiram  Southack,  Aug.  11, 
1862, 12  pairs  congress  gaiters,  at  $2.75 ;  12  pairs  misses'  gaiters, 
at  $1.12;  8  pairs  kip  boots,  at  $2.75;  12  pairs,  at  $.95;  9  pairs 
boys' metallic-toed  shoes,  at  $.72;  12  pairs  gents.' boots,  at  $6.75; 
12  pairs,  at  $4.25;  5  pairs,  at  $3.15.  He  sold  Mr.  Southack  18 
yards  black  silk,  at  $1.17;  48  yards  brown  sheeting,  at  $.19;  18 
yards  crash,  at  $.13,  and  20  yards  flannel,  at  $.45. 
J^^  For  Dictation  Exercises,  see  Key. 


ANALYSIS.  ^i 


ANALYSIS. 

7G,  Analysis  in  arithmetic  consists  in  determining  the  solu- 
tiosn  of  an  example  from  the  relations  of  the  numbers  given  in 
that  example. 

The  given  number  which  is.  of  the  same  denomination  as  the 
required  answer  forms  the  basis  of  all  the  reasoning,  and  should 
be  the  first  written  in  perlbrming  an  example. 

The  value  of  any  number  of  things  may  be  obtained  by  first 
finding  the  value  of  a  single  thing  or  unit  of  the  same  denom- 
ination.    This  unit  is  sometimes  called  the  unit  of  computation. 

Illustrative  Example. 

If  25  barrels  of  flour  cost  $175,  what  cost  17  barrels  ? 
Operatiox.  $175  is  the  term  of  the  same  denom- 

l'<^5        I7::=i^ll9  Ans    ^^^^^^^  ^^  ^^®  required  answer.     Before 
iJ5  finding  the  value  of  17  barrels,  we  must 

know  the  value  of  1  barrel.*  If  25  barrels  cost  $175, 1  barrel  will  cost 
1  twenty-fifth  of  $175,  and  17  barrels  will  cost  17  X  1  twenty-fifth  of 
$175,=:  $119. 

Examples. 

1.  If  13  acres  of  land  produce  780  bushels  of  com,  how  many 
bushels  will  5  acres  produce  ?  -^ns.  300, 

2.  If  5  boxes  of  oranges  cost  $21.80,  what  cost  21  boxes  ? 

Ans.  $91.56. 

3.  If  a  car  runs  207  miles  in  9  hours,  how  far  will  it  run  in 
25  hours  ? 

4.  If  18  rows  of  potatoes  yield  54  bushels,  how  many  bushels 
will  405  similar  rows  yield  ? 

5.  If  $19.74  were  paid  for  14  bushels  of  rye,  what  must  be 
paid  for  25  bushels  ? 

6.  If  19  tons  of  coal  run  an  engine  266  miles,  how  far  will 
14  tons  run  it? 

7.  If  5  oxen  consume  85  pounds  of  hay  in  1  day,  how  much 

will  be  required  for  1  yoke  of  oxen  of  the  same  size,  and  iox  the 

game  time  ? 

*  1  barrel  is  the  unit  of  computation. 


54  ANALYSIS  AND  REVIEW. 

8.  How  many  pounds  of  coffee  can  be  bought  for  $15,  if  40  lbs. 
cost  $8  ? 

Note.  —  If  $8  pay  for  40  pounds,  $1  will  pay  for  1  eighth  of  40 
pounds,  and  $15  will  pay  for  15  X  1  eighth  of  40  pounds  =  75  pounds. 

9.  If  150  barrels  of  apples  were  bought  for  $200  and  sold  for 
$350,  what  would  be  gained  by  selling  45  barrels  at  the  same 
rate  ? 

10.  If  a  quantity  of  hay  lasts  22  oxen  105  days,  how  many 
days  will  it  last  5  yoke  ? 

Note.  — If  it  lasts  22  oxen  105  days,  it  will  last  1  yoke  11  X  105,  and 
it  will  last  5  yoke  1  fifth  of  11  X  105  days  =231  days. 

11.  A  field  of  wheat  was  reaped  by  10  men  in  6  days  ;  what 
length  of  time  would  be  required  for  15  men  to  reap  the  same 
amount  ? 

12.  A  cistern  can  be  emptied  in  35  minutes  by  7  pipes ;  in 
what  time  can  it  be  emptied,  if  5  only  of  the  pipes  are  open  ? 

13.  If  1423  operatives  can  do  a  piece  of  work  in  12  days,  in 
what  time  will  2400  operatives  perform  the  same  work  ? 

14.  If  a  certain  piece  of  work  can  be  performed  by  250  men 
in  14  weeks,  how  many  more  must  be  employed  to  perform  it  in 
a  week  ? 

15.  A  garrison  of  10000  men  have  provision  to  last  them 
6  weeks ;  if  2000  men  be  killed  in  a  sally,  how  long  will  the 
provisions  last  the  remainder? 

77.    Questions  for  Review. 

1.  Federal  Money.  What  are  the  denominations  of  federal 
money?  Give  the  table.  How  do  you  write  numbers  in  federal 
currency  ?  What  is  considered  the  unit  ?  Give  the  sign  for  dollars. 
How  do  you  reduce  eagles  to  dollars  ?  dollars  to  cents  ?  dollars  to 
mills  ?  cents  to  mills  ?  mills  to  dollars  ? 

2.  How  do  you  add  numbers  in  this  currency  ?  How  do  you  sub- 
tract ?  When  you  multiply,  of  what  denomination  is  the  product  ? 
When  you  divide  by  an  abstract  number,  of  what  denomination  is  the 
quotient?  Divide  $185  by  7,  continue  the  division  to  mills,  and 
explain.  What  is  necessary  in  order  to  divide  mills  by  dollars  ?  by 
cents  ?     In  dividing  cents  by  dollars,  is  the  quotient  abstract  or  con- 


REVIEW.  55 

cT3te?     In  dividing  dollars  by  an   abstract  number,  is  the  quotient 
abstract  or  concrete  ? 

3.  Bills.  What  is  a  bill?  Ans.  It  is  a  writing  given  by  the  cred- 
itor to  the  debtor,  showing  the  amount  of  the  debt.  Who  is  the 
creditor  ?  the  debtor  ?     What  is  the  receipt  of  a  bill  ? 

4.  Analysis.  What  is  analysis  ?  ^^^'hich  number  forms  the  basis 
of  the  reasoning. 


T8,    General  Review,  No.  2. 

1.  287  4-  5  million  -f-  36  thousand  -f-  59481  z=  ? 

2.  Add  567  to  the  sum  of  the  following  numbers:  121 ;  232;  343; 
454;  565;  676;  787;  898. 

3.  Take  987  from  each  of  the  following  numbers,  and  add  the  re- 
mainders :    9876 ;  6678  ;  3644  ;  7573 ;  2432  ;  4001. 

4.  What  number  must  be  added  to  the  difference  between  58  and 
7003  to  equal  938425  ? 

5.  What    number,    taken    from    the    quotient    of     1833000-^47 
leaves  25  ? 

6.  What  number  equals  the  product  of  1785,  394,  and  (624—48)  ? 

7.  If  5872  is  the  multiplicand,  and  half  that  number  the  multiplier, 
what  is  the  product? 

8.  If  4832796  is  the  product,  and  1208199  the  multiplicand,  what 
is  the  multiplier  ? 

9.  If  894869  is  the  minuend,  and  the  sum  of  all  the  numbers  in  the 
third  example  is  the  subtrahend,  what  is  the  remainder  ? 

10.  If  700150  is  the  dividend,  and  3685  the  quotient,  what  is  the 
divisor  ? 

11.  If  28936  is  the   divisor,  and  86  is  the  quotient,  what  is  the 
dividend  ? 

12.  Divide  87  million  by  15  thousand. 

13.  $3.75  4- $9.32 +  $.75 +  $10. +$2.185 -}- 4  cents  =:? 

14.  $19.— $.75— $8.25  +  $3.54zzz? 

15.  From  18  X  $5,873,  take  $3.68  +  4. 

16.  If  $183.30  is  the  dividend,  and  $3.90  the  divisor,  what  is  the 
quotient  ? 

17.  If  $98  60  is   the   dividend,  and   17   the  divisor,  what  is   the 
quotient  ? 

J^^  For  changes,  see  Key. 


56  PROPERTIES  OF  NUMBERS. 

PROPERTIES  OF  NUMBERS. 

79.    Signs.  —  Recapitulation. 
-f-  signifies  plus,  or  more.  rr  signifies  equal  to. 

—  signifies  minus,  or  less.  X   signifies  multiplied  by. 

^  signifies  greater  than.  -~-  signifies  divided  hy, 

<^  signifies  less  than.  .•.    signifies  therefore. 

()   parenthesis,    and        ,  vinculum,   signify  that  the  same   op-, 

eration  is  to  be  performed  upon  all  the  quantities  thus 

connected. 

Definitions. 

80.  Numbers  are  either  integral  or  fractional. 

81.  Integral  numbers,  or  Integers,  are  whole  numbers. 
8S.    iPractional  numbers  are  parts  of  whole  numbers. 

83.  A  Factor  or  Divisor  of  a  number  is  any  number 
which  is  contained  in  it  without  a  remainder ;  thus,  2  is  a  fac- 
tor of  6. 

84.  A  Prime  Number  is  a  number  which  contains  no 
integral  factor  but  itself  and  1 ;  as,  1,  2,  3,  11. 

85.  A  Composite  Number  is  a  number  which  contains 
other  integral  factors  besides  itself  and  1 ;  as,  4,  6,  8,  25. 

86.  A  Prime  Factor  is  a  factor  which  is  a  prime  number. 
8T.    A  composite  number  equals  the  product  of  all  its  prime 

factors ;  thus,  12  =  2X2X3. 

88.  Two  numbers  are  said  to  be  prime  to  each  other  when 
they  contain  no  common  factor  except  1 ;  thus,  8  and  15  are 
prime  to  each  other. 

80,  The  Power  of  a  number  is  the  number  itself,  or  the  prod- 
uct obtained  by  taking  that  number  a  number  of  times  as  a  factor. 

The  number  itself  is  the  first  power;  if  it  is  taken  twice  as  a 
factor,  the  product  is  called  the  second  power^  or  square;  if 
three  times,  it  is  called  the  third  power ^  or  cuhe  ;  if  four  times, 
the  fourth  power,  &c.  Thus,  the  second  power  of  3  is  3  X  ^ 
=  9  ;  the  third  power  of  3  is  3x3X3  =  27;  the  fiflh 
power  of  3  is  3X3X3X3X3  =  243. 


DIVISIBILITY  OF  NUMBERS.  57 

90,  Tlie  Index  or  Exponent  of  a  power  is  a  figure  which 
shows  how  many  times  the  number  is  taken  as  a  factor.  It  is 
written  at  the  right  of  the  number,  and  above  the  line.  Thus, 
in  5^,  7^,  2'*,  the  exponent  ^  shows  that  5  is  taken  three  times  as 
a  factor,  ~  that  7  is  taken  twice,  and  4  that  2  is  taken  four  times 
as  a  factor. 

©1,  The  Root  of  a  number  is  one  of  the  equal  factors 
which  produce  that  number.  If  it  is  one  of  the  two  equal 
factors,  it  is  the  second,  or  square  root ;  if  one  of  the  three,  the 
third,  or  cube  root;  if  one  of  the  four,  the  fourth  root,  &c. 
Thus  the  square  root  of  9  is  3,  the  cube  root  of  125  is  5. 

9^.  V  ^s  the  Radical  Sign,  and,  by  itself,  denotes  the 
square  root ;  with  a  figure  placed  above,  it  denotes  the  root  of 
that  degree  indicated  by  the  figure  ;  thus,  ^  signifies  the  third 
root,  -^  the  sixth  root. 

Divisibility  of  Numbeks. 

93,  (1.)  Any  number  whose  unit  figure  is  0,  2,  4,  6,  or  8, 
is  even. 

(2.)   Any  number  whose  unit  figure  is  1,  3,  5,  7,  or  9,  is  odd. 

(3.)  Any  even  number  is  divisible  by  2. 

(4.)  Any  number  is  divisible  by  3  when  the  sum  of  its  digits 
is  divisible  by  3  ;  thus,  2814  is  divisible  by  3,  for  2  +  8  +  1  +  4 
p=.  15,  is  divisible  by  3. 

(5.)  Any  number  is  divisible  by  4,  when  its  tens  and  units 
are  divisible  by  4 ;  for,  as  1  hundred,  and  consequently  any  num- 
ber of  hundreds,  is  divisible  by  4,  the  divisibility  of  the  given 
number  by  4  must  depend  upon  the  tens  and  units  ;  thus,  86324 
is  divisible  by  4,  while  6831  is  not. 

(G.)  Any  number  is  divisible  by  5  if  the  units*  figure  is  either 
S  or  0 ;  for,  as  1  ten,  and  consequently  any  number  of  tens,  is 
divisible  by  5,  the  divisibility  of  the  given  number  by  5  must 
depend  upon  the  units. 

(7.)  Any  number  is  divisible  by  6,  if  divisible  by  3  and  by  2. 

(8.)  Any  number  is  divisible  by  8,  if  its  hundreds,  tens,  and 
enits  are  divisible  by  8 ;  for,  as  1  thousand,  and  consequently  any 


58  I'HOPERTIES  OF  NUilBEIiS. 

number  of  tliousands  is  divisible  by  8,  tlie  divisibility  of  the  given 
number  by  8  must  depend  on  the  hundreds,  tens,  and  units. 

(9.)  Any  number  is  divisible  by  9  if  the  sum  of  its  digits  is 
divisible  by  9*;  thus,  368451  is  divisible  by  9,  and  23476  is  not. 

(10.)  Any  number  is  divisible  by  10,  100,  or  1000,  if  it  con- 
tain at  the  right  1,  2,  or  3  zeros  ;  and  so  on. 

(11.)  Any  number  is  divisible  by  11,  if  the  difference  between 
the  sums  of  the  alternate  digits  is  0,  or  a  number  divisible  by  11 ; 
thus,  in  126896,  as  (1  +  6  +  9)  —  (2  -f-8  +  6)  =:  0,  the  number 
is  divisible  by  11  ;  and  in  9053,  as  (9  +  5)  —  (0  +  3)  z=  11,  the 
number  is  divisible  by  11. 

(12.)  A  number  is  divisible  by  any  composite  number,  if  it  is 
divisible  by  all  the  factors  of  that  number. 

03,  There  are  no  rules  of  sufficient  practical  importance  for 
determining  when  numbers  are  divisible  by  other  numbers  than 
those  spoken  of  above.  Their  divisibility  must  be  ascertained 
by  trial.     To  do  this, — 

Divide  the  number  successively  hy  higher  and  higher  primes, 
until  one  is  found  which  divides  it,  or  until  the  quotient  is  smcdler 
than  the  divisor.  If  no  divisor  is  then  found,  the  number  is 
prime;  for,  if  a  number  contain  any  prime  factor  greater  than 
its  square  root,  its  corresponding  factor  must  be  less. 

94.  If  the  odd  numbers  are  written  in  order,  and  every 
third  one  from  3,  every  fifth  one  from  5,  every  seventh  one  from 
7,  and  so  on,  be  marked,  and  the  figures  3,  5,  7,  &c.,  be  written 
under  the  figures  as  they  are  marked,  the  remaining  numbers 
will  be  primes,  and  those  marked  will  have  for  their  factors  the 
numbers  written  beneath ;  f  thus,  — 
1,  3,  5,  7,  0,  11,  13,  X$,  17,  19,  M,  23,  ^^,  ^t,  29,  31, 

3  3,5  3,7  5        3,9 

0;gl,  ^$,  37,  210,  41,  43,^^,  47,  ^0,  $X,  &c. 

3,11    6,7  3,13  3,5  7      3,17 

9,16 

*   See  Appendix. 

f  Eratosthenes,  in  the  third  century  B.  C,  discovered  this  method  of 
finding  primes  and  factors  of  numbers,  and  as  he  made  his  table  of  parch- 
ment, cutting  out  the  composite  numbers  as  he  found  them,  this  parchment 
wai  called  Eratosthenes'  Hiece. 


TABLES   OF  PRIME  AND   COMPOSITE  NUMBERS. 


59 


By  applying  this  principle,  a  tabic  can  easily  be  made  of  the 
primes  and  of  the  composites,  with  their  factors. 

Table  of  Pkimk  Nu.mbkiis  to   1201. 


I 

(51 

151 

251 

359 

403 

593 

701 

827 

953 

1069 

2 

GZ 

157 

257 

3G7^ 

4G7 

599 

709 

829 

967 

1087 

3 

71 

1()3 

2(53 

373 

479 

GOl 

719 

839 

971 

1091 

5 

73 

1G7 

2(>9 

379 

487 

G07 

727 

853 

977 

1093 

7 

79 

173 

271 

383 

491 

G13 

733 

857 

983 

1097 

11 

83 

179 

277 

389 

499 

G17 

739 

859 

991 

1103 

13 

89 

181 

281 

397 

603 

G19 

743 

863 

997 

1109 

17 

97 

191 

283 

401 

509 

G31 

751 

877 

1009 

1117 

19 

101 

193 

293 

409 

621 

G41 

757 

881 

1013 

1123 

23 

103 

197 

307 

419 

523 

643 

761 

883 

1019 

1129 

29 

107 

199 

311 

421 

541 

647 

769 

887 

1021 

1151 

31 

109 

211 

313 

431 

547 

G53 

773 

907 

1031 

1153 

37 

113 

223 

317 

433 

657 

659 

787 

911 

1033 

1163 

41 

127 

227 

331 

439 

5G3 

661 

797 

919 

1039 

1171 

43 

131 

229 

337 

443 

5G9 

673 

809 

929 

1049 

1181 

47 

137 

233 

347 

449 

671 

677 

811 

937 

1051 

1187 

53 

139 

239 

349 

457 

577 

683 

821 

941 

1061 

1193 

59 

149 

241 

353 

461 

587 

691 

823 

947 

1063 

1201 

Table  of  the  Composite  Numbers  to  917, 
Which  contain  no  prime  factor  less  than  7   {excepting  1*). 


Nos. 

Factors. 

Nos. 

Factors. 

Nos.  Factors. 

Nos. 

Factors. 

Nos. 

Factors. 

49 

72 

289 

17« 

469   7,  67 

623 

7,89 

779 

19,41 

77 

7,  11 

299 

13,  23 

473  11,  43 

629 

17,  37 

781 

11,71 

91 

7,  13 

301 

7,  43 

481  13,  37 

637 

7',  13 

791 

7,  113 

il9 

7,  17 

319 

11,  29 

493  17,  29 

649 

11,  59 

793 

13,  61 

121 

112 

323 

17,  19 

497   7,  71 

667 

23,  29. 

799 

17,47 

133 

7,  19 

329 

7,  47 

511   7,  73 

671 

11,  61 

803 

11,  73 

143 

11,  13 

341 

11,  31 

517  11,  47 

679 

7,  97 

817 

19,  43 

161 

7,  23 

343 

7^ 

527  17,  31 

689 

13,  53 

833 

72,  17 

169 

13« 

361 

192 

529  23'<» 

697 

17,  41 

841 

292 

187 

11,  17 

371 

7,53 

533  13,  41 

703 

19,  37 

847 

7,  11» 

203 

7,  29 

377 

13,  29 

539  7^  11 

707 

7,  101 

851 

23,  37 

209 

l.l,  19 

391 

17,  23 

551  19,  29 

713 

23,  31 

869 

11,  79 

217 

7,  31 

403 

13,  31 

553   7,  79 

721 

7,  103 

889 

7,  127 

221 

13,  17 

407 

11,  37 

559  13,  43 

731 

17,  43 

893 

19,  47 

247 

13,  19 

413 

7,  59 

581   7,  83 

737 

11,  67 

899 

29,  31 

253 

11,  23 

427 

7,  61 

583  11,53 

749 

7,  107 

901 

17,  53 

259 

7,37 

437 

19,  23 

589  19,  31 

763 

7,  109 

913 

11,  83 

287 

7,  41 

451 

11,  41 

611  13,  47 

767 

13,  59 

917 

7,  131 

*  1  is  a  factor  of  all  nTimbers. 


60 


PROrERTIES  OF  NUMBERS. 


FACTORING  OF  NUMBERS. 
Oo.    Illustrative  Example,  I. 
Resclve  48  into  its  prime  factors. 

OrERATION. 

48  =6X8;  6  =  2X3;  8  =  2X2X2;.-.  48  =  2X 
2  X  2  X  2  X  3,  or  2^*  X  3.     Hence, 

Rule  I.  To  resolve  a  number  into  its  prime  factors. — • 
First  separate  it  into  any  tivo  factors  ;  separate  these  factors,  if 
they  are  composite,  into  others,  and  so  on,  till  all  are  prime. 

Proof.  Multiply  the  factors  thus  obtained  together,  and  the 
product,  if  the  work  is  correct,  will  equal  the  given  number. 

96.   Examples. 
Resolve  the  following  numbers  into  their  prime  factors.  — 
1.     32. 


Ans.  25. 

2.  84. 
Ans.  22  X  3  X  7. 

3.  88. 
Ans.  23  X  11. 


4.     56. 

7. 

100. 

5.    49. 

8. 

150. 

6.     72. 

9 

69. 

10. 

11. 

12. 


81. 
99. 
144. 


13. 
14. 
15. 


64. 

77. 
108. 


16. 

130 

17. 

125. 

18. 

250a 

07o    Illustrative  Example,  II. 
Resolve  42075  into  its  prime  factors. 


Operation. 
3 ) 42075 

3 ) 14025 

5)4675 


Here  we  divide,  successively,  by  such  prime 
numbers  as  will  leave  no  remainder,  till  we  obtain 
a  prime  number  for  a  quotient ;  since  the  product, 
of  these  prime  numbers,  3,  3,  5,  5^  11,  and  17  equals 
the  given  number,  they  must  be  the  prime  factors 
of  that  number.     Hence, 


5)935 

11)187 

17 
iw5.  32  52  11, 17. 

Rule  IL  Divide  th.  number  by  any  prime  number  which  is 
^ntained  in  it  without  a  remainder.  Divide  the  quotient  in  the 
same  manner,  and  thus  continue  till  a  quotient  is  obtained  which 
is  a  prime  number.  This  quotient  and  the  several  divisors  are  the 
^rime  factor:-. 


GllEATEST  COMMON  DIVISOR. 


61 


Note.  —  Tlie  work  may  sometimes  be  shortened  by  dividing  by  a  com- 
posite number,  remembering  afterwards  to  substitute  the  factors  of  that 
number  for  the  number  itself.  Thus,  in  the  above  we  may  divide  by 
9  instead  of  dividing  by  3  twice. 

98,   Examples. 
Resolve  the  following  numbers  into  their  prime  factors. 

19.  17G.  A71S.  2^  X  11. 

20.  180.  Ans.  2^  X  3^  X   5. 

21.  192.  Ans.  2«  X  3. 

22.  208.  Ans.  2^  X  13. 

00,  Select  the  prime  numbers  in  the  columns  below,  and 
find  the  factors  of  the  composite  numbers. 


23. 

260. 

27. 

357. 

24. 

285. 

28. 

644. 

25. 

329. 

29. 

684. 

26. 

338. 

30. 

2310. 

1. 

341. 

6. 

450. 

11. 

704. 

16. 

947. 

2. 

344. 

7. 

590. 

12. 

719. 

17. 

971. 

3. 

362. 

8. 

560. 

13. 

769. 

18. 

2681. 

4. 

367. 

9. 

596. 

14. 

808. 

19. 

1163. 

5. 

409. 

10. 

689. 

15. 

839. 

20. 

3248. 

J^*  Por  Dictation  Exercises,  see  Key. 

Greatest  Common  Divisor. 

100.  A  L-ommon  Divisor  of  two  or  more  numbers  is  any 
number  that  will  exactly  divide  each  of  them ;  thus,  ^  is  a 
common  divisor  of  12  and  18. 

101.  The  Greatest  Coinmoii  Divisor  is  the  greatest  num- 
ber that  will  exactly  divide  each  of  them ;  thus,  6  is  the  great- 
est common  divisor  of  12  and  18. 

103,   Illustrative  Example. 

Find  the  greatest  common  divisor  of  12,  30,  and  42. 

Operation. 
-^ oyoyq  As  2  and  3  are  the  only  common  fac- 

QQ 2  X  3  X  ■'*  ^^^^    ^^   "^^'  ^^'  ^^^   ^^'  ^^  follows  that 

^2 2X3x7*  2  X  3,  or  6,  is  the  greatest  common  di- 

G  C.  D.  ^  2  X  3  zir  ^Ans.    ^'^''^'     H^^^^' 


Rule  I.     To  find   the  greatest  common  divisor  of  two  or 


(52  GREATEST  COMMON  DIVISOR. 

more  numbers :  Separate  the  numhers  into  their  prime  facton^ 
and  jind  the  product  of  such  as  are  common. 

J  103.    Examples. 

Find  the  G.  C.  D.*  of 


1.  48,  56,  and  60.      Atis.  4. 

2.  24,  42,  and  54.      Ans.  6. 


3.  108,  45,  18,  and  63. 

4.  18,36,  12,  48,  and  42 


Note.  —  In  Example  4,  18  is  a  factor  of  36,  and  12  of  48.  The  G.C.  D. 
of  18  and  12  must  be  the  G.  C.  D.  of  18,  12,  and  their  multiples,  36  and 
48  ;  .-.  we  need  only  find  the  G.  C.  D.  of  18,  12,  and  42. 

Find  the  G.  C.  D.  of 


5.  42,  28,  and  84. 

6.  26,  52,  and  65. 


7.  32,  18,  108,  and  25. 

8.  114,  102,  78,  and  66. 


104:.  When  numbers  cannot  readily  be  separated  into  their 
factors,  the  following  method  may  be  adopted :  — 

Illustrative  Example.     Find  the  G.  C.  D.  of  91  and  325. 

Operation.  We  divide  325  by  91,  to  see  if  it  is  a 

9A  )  325  (  3  divisor  of  325,  for  91  is  the  greatest  divisor 

273  of  itself;  if  it  is  a  divisor  of  325,  it  is  the 

52  )  91  (  1  G.  C.  D.  of  91  and  325.     It  is  not  a  divisor 

52  of  325,  for  there  is  a  remainder  of  52.     52 

39  ^  62(\  ^^  ^^^  greatest  divisor  of  itself;    if  it  is  a 

39  divisor  of  91,  it  is  the  G.  C.  B.  of  52  and 

~V\\  oq  /  q    ^1'     It  is  not  a  divisor  of  91,  for  there  is  a 

OQ         remainder  of  39  ;  39  is  the  greatest  divisor 

of  itself;  if  it  is  a  divisor  of  52,  it  is  the 

^^  G.  C.  D.  of  39  and  52.     It  is  not  a  divisor  of 

52,  for  there  is  a  remainder  of  13 ;  13  is  the  G.  C.  D.  of  itself  and  39. 
It  must  therefore  be  of  39  and  52,  for  52  :=  1  X  39  +  13.  If  it  is  the 
G.  C.  D.  of  39  and  52,  it  must  be  of  52  and  91,  for  91  z=l  X  52 + 
39.  If  it  is  the  G.  C.  D.  of  52  and  91,  it  must  be  of  91  and  325,  for 
325  =  3  X  91  -f  52.     Hence  the  following : 

Rule  IT.  To  find  the  G.  C.  D.  of  two  numbers:  Divide 
the  greater  number  hy  the  less,  and  the  less  number  by  the  re- 
mainder,  if  there   is   any,  and  thus  proceed,   dividing  the  last 

*  Greatest  Common  Divisor, 


GREATEST  COMMON  DIVISOR.  G3 

livisor  hy  the  last  remainder,  until  nothing  remains.      The  last 
iivisor  is  the  G.  O.  I),  sought. 

To  find  the  G.  C.  D.  of  more  than  two  numbers,  Jind  the 
G.  C.  D.  of  ariy  two  of  them,  and  then  of  that  divisor  and  a 
third  number,  and  so  on  till  all  the  numhej's  are  taken. 

10«5«    Examples. 
Find  the  G.  C.  D.  of 


12.  229  and  954. 

13.  392,  1008,  and  224. 

14.  6581,  6611,  and  249. 


9.    198  and  297.      Ans.  99. 

10.  222  and  564.        Aiis.  6. 

11.  529,  782,  and  1127. 

Ans.  23. 

15.  What  is  the  width  of  the  widest  carpeting  that  will  ex- 
actly fit  either  of  two  halls,  45  feet  and  33  feet  wide  respect- 
ively ?  Ans.  3  ft. 

16.  A  has  a  piece  of  ground  90  feet  long  and  42  feet  wide; 
what  is  the  length  of  the  longest  rails  that  will  exactly  suit  its 
length  and  its  width  ?  Ans.  6  ft. 

17.  A  lady  has  one  flower  bed  measuring  10  feet  around,  and 
another  measuring  18  feet.  If  she  borders  the  beds  with  pinks, 
what  is  the  greatest  distance  she  can  set  her  pink  roots  apart, 
and  have  them  equally  distant  in  the  two  beds  ?  Ans.  2  ft. 

18.  A  man  has  90  bushels  Kidney  potatoes,  60  bushels  Jack- 
son Whites,  and  105  bushels  Red  Rileys.  If  he  puts  them  all 
into  the  largest  bins  of  equal  size  that  will  exactly  measure  either 
lot,  how  many  bushels  will  each  of  his  bins  contain  ? 

19.  What  is  the  length  of  the  longest  stepping-stones  that  will 
exactly  fit  3  streets,  72,  51,  and  87  feet  wide,  respectively  ? 

20.  What  is  tlve  length  of  the  longest  curb-stones  that  will 
exactly  fit  4  strips  of  sidewalk,  the  first  being  273  feet  long,  the 
*^cond  294,  the  third  567,  and  the  fourth  651  ? 

1^  For  Dictation  Exercises,  see  Key, 

A 


64  COMMON  FRACTIONS. 


FRACTIONS. 

1^6*  A  Fraction  is  one  or  more  of  the  equal  parts  of  a 
unit;  thus,  f,  read  three  fourths,  shows  that  a  unit  has  been 
divided  into  four  equal  parts,  and  that  three  of  those  parts  are 
taken. 

lOT,  The  number  which  shows  into  how  many  equal  parts  \ 
unit  is  divided,  is  called  the  Denominator  of  the  fraction, 
because  it  denominates  or  names  the  parts  ;  thus,  4  is  the  denom- 
inator of  f . 

108,  The  number  which  shows  how  many  parts  are  taken,  is 
called  the  Numerator ;  thus,  3  is  the  numerator  of  f . 

109,  The  numerator  and  denominator  are  called  the  Terms 
of  a  fraction. 

110,  A  Common  or  Vulgar  Fraction  is  a  fraction  whose 
denominator  and  numerator  are  both  expressed,  the  numerator 
being  written  above,  and  the  denominator  below,  a  dividing 
line  ;  as,  ^,  |,  /^. 

IfiS.  A  Decimal  Fraction  is  one  whose  denominator  is  10, 
<!»•  some  integral  power  of  10.  The  denominator  is  not  gen- 
erally expressed,  -f^  written  .2,  and  -^^^  written  .36,  are  decimal 
fractions. 

11^.  A  Mixed  Number  is  a  w^hole  number  and  a  fraction 
expressed  together,  as  7|,  2 If. 

ll«l.  Common  fractions  may  be  either  Proper,  Improper, 
Compound,  or  Complex. 

1 14.  A  Proper  Fraction  is  one  whose  numerator  is  less  than 
its  denominator,  as  f . 

11^«  An  Improper  Fraction  is  one  whose  numerator  equals 
or  exceeds  lis,  denominator,  as  §f,  |-. 

116,  A  Compound  Fraction  is  a  fraction  of  a  fraction,  as  | 

off 

117,  A  Complex  Fraction  is  one  which  contains  a  fraction 
In  either  or  both  of  its  terms,  as  2f ,   f 

"7  ^' 


GENERAL  PRINCIPLES.  C5 

Questions.  What  does  the  denominator  of  a  fraction  show? 
What  does  the  numerator  show  ? 

What  is  meant  by  the  expression  f  ?  Ans,  5  of  the  8  equal 
parts  into  which  a  unit  is  divided. 

What  is  meant  by  the  expression  |  ?  ^\?  ^§  ?  ^-f  ? 

118.  If  we  compare  common  fractions  with  the  last  expression 
for  division  in  Art.  55,  we  shall  see  that  their  forms  are  alike. 
A  fraction  implies  division,  the  numerator  being  the  dividend, 
and  the  denominator  the  divisor.  Thus,  f  may  be  considered 
either  three  fourths  of  1  or  one  fourth  of  3.  The  following 
diagram  will  show  that  these  are  equivalent  expressions,  f  of 
the  one  line  in  figure  1  being  equal  to  ^  of  the  three  lines  in 
figure  2. 

Fig.  1.  Fig.  2. 

I 1 ! — ! ! 

HH — M — 1  ^-H — f-H — I 

i — I — I — I — I 

f  of  1  ==  f .  i-  of  3  =  J. 

What  does  ^  denote  ?   Ans.     It  denotes  either  5  of  the  seven 
equal  parts  into  which  1  is  divided,  or  one  seventh  of  5. 
What  does  ^\  show  ?  f  f  ?  if  ? 

119,     General  Principles. 

Note.  —  The  following  propositions  should  be  copiously  illustrated  by 
the  teacher,  and  frequently  referred  to,  until  they  are  fully  comprehended 
by  the  pupil. 

Proposition  I.  As  the  denominator  of  a  fraction  shows  the 
number  of  parts  into  which  a  unit  is  divided,  and  the  numerator 
shows  how  many  parts  are  taken,  it  follows  that  if  we  multiply 
the  numerator  of  a  fraction  by  a  whole  number,  we  multiply  the 
number  of  parts,  and  thus  iricrease  the  value  of  the  fraction;  but 
if  we  multiply  the  denominator  of  a  fraction,  we  multiply  the 
number  of  parts  into  which  a  unit  is  divided,  and  thus  diminish 
the  size  of  the  parts,  and  consequently  decrease  the  value  of 
the  fraction. 

5 


66  COMMON  FBACTIOtrs. 


1 1 \ \ i — +-M 

ILLUSTRA-  2X2  4  I  1  I  ! r  (      -n,       ..       . 

TiON.       — T^ —  =  -  r^ 1     Praction  increased, 

2             2  > 

5  X  2^^^^  10   r^ |"™|j 1 ' — I ^ ! — [      Fraction  diminished. 

Proposition  II.  If  we  divide  the  nmnerator  of  a  fraction  bt 
a  whole  number,  we  divide  the  number  of  parts  and  thus  diminish 
the  value  of  the  fraction  ;  but  if  we  divide  the  denominator  of  a 
fraction,  we  divide  the  number  which  shows  into  how  many  parts 
the  unit  is  divided,  and  thus  increase  the  size  of  the  parts,  and 
consequently  increase  the  value  of  the  fraction. 

^  I    I   M   I   H 

2-^2 1 

Illustration,  "l      — 6    |— ( — j 1 — | — } — |     Fraction  diminisx^ed. 

2             2 
g  _^  2  ~— -  3    h"n^ 1 '      Fraction  increased. 

Proposition  III.  If  we  multiply  the  numerator  and  denom- 
inator, each  by  the  same  number,  we  increase  the  number  of 
parts  of  the  fraction,  but  diminish  their  size  in  the  same  propor- 
tion ;  consequently  the  value  of  the  fraction  is  not  altered. 

Q 

ILLUSTRATION         *  Fraction  not  altered 

?J<i_6          t     ,     ,     ,     ,     ,     ,  ,  in  value. 

4X2  —  8  I  ill 1 

Proposition  IV.  If  we  divide  the  numerator  and  denom- 
inator, each  by  the  same  number,  we  diminish  the  number  of 
•parts  in  the  same  proportion  as  we  increase  ^  their  size,  conse- 
quently the  value  of  the  fraction  is  not  altered. 

:-  [1111 — H 

Illustration.  Fraction  not  altered 

4-^2 2  I  T  I j  in  value. 

Questions.  How  does  multiplying  the  numerator  of  a  frac- 
tion affect  the  value  of  the  fraction  ?  Why  ?  How  does  multi- 
plying the  denominator  affect  the  value  of  the  fraction  ?     Why  ? 

How  does  dividing  the  numerator  of  a  fraction  affect  the  value 


Deduction. 


67^ 


of  the  fraction  ?  Why  ?  How  does  dividing  the  denominator 
ffect  the  value  of  the  fraction  ?     Why  ? 

If,  then,  you  multiply  the  numerator  and  denominator  each 
uy  the  same  number,  what  is  the  effect  upon  the  fraction? 
Why? 

If  you  divide  the  numerator  and  denominator  each  by  the 
mme  number,  what  is  the  effect  upon  the  fraction  ?     Why  ? 

Reduction  of  Fractions  to  Lovtest  Terms. 

1^0.  A  Fraction  is  expressed  in  its  lowest  terms  when  the 
numerator  and  denominator  are  prime  to  each  other. 

131,    III.  Ex.     Reduce  -^^  to  its  lowest  terms. 

Operation.  8==4X2;10i=:5X2.     Dividing  the 


8       4X^ 


numerator  and  denominator  each  by  strik- 
ing out  the  common  factor  2,  the  value  of 
the  fraction  will  not  be  altered  (Art.  119, 


10      5X^      5 

Prop.  IV.),  and  will  T)e  expressed  in  its  lowest  terms.    Hence  the 

Rule.     To  reduce  a  fraction  to  its  lowest    terms  :    Remove 
from  the  numerator  and  denominator  all  their  common  factors, 

1^3.     Examples. 
Reduce  to  their  lo-west  terms, 


if.    ^ns.^. 


.      f|.   Ans.^-,.  7.  ffg.  10.     ^\S 

2.  e.    Ans,^.       5.    ^%%,  8.  ^\\.  11.     U%' 

3.  tf.    Ans.l.       6.      if.  9.  m-  12.     fM- 

13.   Reduce  Jf  to  its  lowest  terms.  Ans.  ^, 

Note  I.  —  1,  being  a  factor  of  all  numbers,  will  remain  when  all  other 
factors  are  struck  out,  as  in  the  numerator  of  example  13. 

Note  II.  —  In  case  the  factors  of  the  numerator  and  denominator  can- 
not readily  be  ascertained, /wrf  the  G.C.D.of  the  two  tertns,  and  divide 
each  of  them  by  it. 

Reduce  to  their  lowest  terms, 


14. 
15. 


i/A. 


16. 
17. 


18. 
19. 


7  37 


-ES 


20. 
21. 


|^°  For  Dictation  Exercises,  see  Key. 


68  COMMON  FRACTIONS. 

Cancellation. 
1^3.    Cancellation  consists  in  rejecting  equal  factors  from 
dividend  and  divisor. 

1^4*  All  arithmetical  operations  in  division  may  be  ex- 
pressed in  the  form  of  a  fraction,  the  dividend  being  the  numer- 
ator, and  the  divisor  the  denominator ;  since  dividing  both  terms 
of  a  fraction  by  the  same  number  does  not  alter  its  value,  it  fol- 
lows, that  we  may  strike  out,  or  cancel,  any  factors  common  to  the 
dividend  and  divisor  without  changing  their  relative  value. 

N.  B.  All  operations  wpon  arithmetical  quantities  should  first 
he  expressed,  as  far  as  possible,  by  signs,  that  the  processes  may  he 
clearly  indicated  to  the  teacher,  and  that  the  work  to  he  done  may 
he  reduced,  if  possible,  by  cancellation. 

III.  Ex.  Divide  3  times  4  times  6  times  5  times  7,  by  2 
times  8  times  6  times  9  times  10. 

0X^X0XgX  7  _  7  _  1_  ^^^ 

2X^X0X0X^0        2X2X3X2        24* 
2  3        2 

13^.    Examples. 
Express,  cancel,  and  perform  the  following :  — 

1.  Divide  8X6X3X9X7X4,  by  2X5X7X  10  X  8. 

Ans.  6^f. 

2.  Divide  81  X  42,  by  99  X  7. 

3.  Multiply  75  X  10,  by  3  X  6,  and  divide  that  product  by 
15  X  25  X  12. 

4.  Divide  7  X  8  X  48,  by  63  X  4  X  5  X  17,  and  multiply 
the  quotient  by  51. 

5.  Divide  99  X  28  X  6,  by  5  X  8  X  18,  multiply  the  quo- 
tient by  4  X  4,  and  divide  by  22  X  27. 

6.  Spent  ^  of  $75,  which  I  received  for  work,  for  flour  at  $5 
a  barrel ;  how  many  barrels  did  I  buy  ? 

7.  If  25  pounds  of  lead  costs  $4.60,  what  do  8  pounds  cost? 

8.  "What  will  be  received  for  27  pieces  of  broadcloth,  if  6 
pieces  bring  $864? 


CANCELLATION.  69 

.  9.  If  it  requires  13  bushels  of  wheai;  to  make  3  barrels  of 
tour,  how  many  bushels  will  be  required  to  make  78  barrels  of 
lour  ?  Am,  338  bushels. 

10.  If  a  tree  69  feet  high  casts  a  shadow  of  90  feet,  what 
length  of  shadow  will  be  cast  by  a  tree  92  feet  high  ? 

Ans.  120  feet. 

11.  A  merchant  exchanged  561  pounds  of  sugar,  at  9  cents 
)er  pound,  for  eggs  at  11  cents  per  dozen;  how  many  dozen  were 
•eceived  ? 

12.  If  12  pieces  of  cloth,  each  piece  containing  62  yards,  cost 
$372,  what  cost  24  yards  ? 

13.  If  a  bar  of  iron  8  feet  long  weighs  36  pounds,  what  will  a- 
bar  of  the  same  size  100  feet  long  weigh? 

14.  How  many  boxes  of  oranges  can  be  bought  for  $420,  if 
$28  be  paid  for  7  boxes  ? 

15.  If  the  work  of  7  men  is  equal  to  the  work  of  9  boys,  ho^^ 
many  men's  work  will  equal  the  work  of  63  boys  ? 

16.  If  15  men  consume  a  barrel  of  flour  in  6  weeks,  how  lon^ 
would  it  last  9  men  ?  Ans.  10  weeks. 

17.*  If  the  interest  of  $650  for  12  months  is  $52,  what  is  the 
interest  of  three  times  that  sum  for  eight  months?       Ans.  $104. 

18.  If  12  men  can  build  a  wall  in  42  days,  how  many  days 
will  be  required  for  21  men  to  build  it? 

19.  If  $15  purchase  12  yards  of  cloth,  Iioav  many  yards  will 
$48  purchase  ?  Ans.  38f  yards. 

20.  A  ship  has  provision  for  15  men  12  months ;  how  long 
will  it  last  45  men  ? 

21.  How  many  overcoats,  each  containing  4  yards,  can  be 
made  from  10  bales  of  cloth,  12  pieces  each,  42  yards  in  each 
piece  ? 

22.*  If  375  barrels  of  pork,  each  200  pounds,  cost  $6000,  what 
is  the  cost  of  5  barrels,  each  195  pounds  ? 

23.*  Sold  20  barrels  of  apples  at  $2.50  per  barrel,  and  spent 
the  money  thus  obtained  for  cloth  at  $.50  a  yard,  which  I  sold  at 
$.60  a  yard,  and  bought  a  horse  with  the  proceeds.  What  did  I 
pay  for  the  horse? 


yO  COMMON  FRACTIONS. 

196*     Reduction  of  Whole  or  Mixed  Numbers  to 
Improper  Fractions. 

III.  Ex.     Change  2|-  to  an  improper  fraction. 

Operation.  In  1  unit  there  are  |,  .'.  in  2  units 

^i  =  —8  —  ^^^~*  ^^*-      there  are  2  X  |  or  -i/.   \^  -f  |  =  Y,    Ans, 
Hence  the 

Rule.  To  reduce  a  mixed  number  to  an  improper  fraction ;  — 
Multiply  the  whole  number  hy  the  denominator  of  the  fraction  ;  to 
that  product  add  the  numerator ;  and  write  the  result  over  the 
denominator. 

Examples. 

Reduce  to  improper  fractions, 


1.  2j.     Ans.  -V-. 

2.  ^.     Ans.    f. 


3.  7f.     I     5.     llif. 

4.  ^.     I     6.     14f^. 


7.  321f. 

8.  8||t. 


9.    Reduce  36  to  fifths.     ^'  z=  '-?,  ^ns. 

10.  Reduce  584f,  368,  87|,  to  ninths. 

11.  Add  784  to  916,  and  express  the  answer  in  sevenths. 

12.  Reduce  7  X  98  to  eighths. 

13.  Reduce  (15  —  8)  X  16  to  fifths. 

14.  Reduce  8692  to  a  fraction  whose  denominator  is  25. 

15.  Reduce  367^f  to  an  improper  fraction. 

16.  Change  4567f  to  ninths,  43862  to  elevenths. 

17.  Change  36f§j-  to  an  improper  fraction. 
B^  For  Dictation  Exercises,  see  Key. 

137*     Reduction   op  Improper    Fractions    to  Whole 
OR  Mixed  Numbers. 

III.  Ex.     Change  %^-  to  a  mixed  number. 
Operation.  There   are  f  in  1   unit,  .'.in  ^^  there  are  as 

^  ^^^  4|>  A^is.      many  units  as  |  is  contained  times  in  ^g^,  which  is  4|- 
times.    Hence  the 

Rule.     To  reduce  an  improper  fraction  to  a  mixed  number;  — 
Divide  the  numerator  hy  the  denominator. 


MULTIPLICATION  OF    FRACTIONS. 


71 


Examples. 
Change  to  whole  or  mixed  numbers, 


9_8 
"5  • 


Alls.  4:^. 
Ans.  211. 
Ans.  19^. 


6. 

7. 

8. 

9. 

10. 


865 

■Er2T- 

3  7J)_4  8, 


11. 

12. 
13. 
14. 
15. 


J96 

F8- 


1^"  For  Dictation  Exercises,  see  Key. 

A  138.     Multiplication  of  Fractions  by  Whole 
Numbers. 

As  multiplying  the  numerator  of  a  fraction  multiplies  the 
number  of  parts,  their  size  remaining  the  same,  and  dividing  the 
denominator  multiplies  the  size  of  the  parts,  their  number'  re- 
maining the  same  (Art.  119),  it  follows  that, — 

To  multiply  a  fraction  by  a  w^hole  number,  we  may  either 
multiply  the  numerator  by  the  whole  number,  or  divide  the  denom- 
inator. 

The  latter  method  is  preferable  when  the  denominator  can  be 
divided  without  a  remainder,  as  it  gives  the  answer  in  lower 
terms. 

III.  Ex.     Multiply  |  by  4.  , 


1st  Operation. 
7x4        28_„,      . 
- —  =  -  =  3|,  Ans. 


2D  Operation. 

=  ~z=z3h  Ans. 


8  °'  8-T-4         2 

"We  might  have  cancelled  in  the  first  operation,  and  thus  have 


$         ' 

Multiply 

Examples. 

2 

1. 

i  by     5. 

Atis.  2f. 

9. 

^\ 

by       19. 

2. 

3^  by      6. 

Ans.  ^|. 

10. 

t\\ 

by       21. 

3. 

^^   by  504. 

Ans. 

23fi. 

11. 

T%h 

by       95. 

4. 

r\   by     15- 

12. 

A^ 

by       54. 

5. 

f   by       4. 

13. 

m 

by     274. 

6. 

A   by  110. 

14. 

■b\% 

by     328. 

7. 

tVj  by      9. 

15. 

^fk 

by     762. 

8. 

U   by     11. 

16. 

_33^8^7JL 

by       55. 

72  COMMON  FRACTIONS. 

17.  If  one  yard  of  cloth  co^ts  |  of  a  dollar,  what  will  17 
yards  cost  ? 

18.  If  a  ton  of  coal  costs  |  of  an  Eagle,  how  much  will  15 
tons  cost? 

19.  Required  the  cost  of  28  pounds  of  candles,  at  f  of  a  dol- 
lar a  pound. 

20.  Multiply  25G|  by  18.  Ans.  4623f. 
"  21.   Multiply  376ff  by  21. 

^^  For  Dictation  Exercises,  see  Key. 

1S9*   Multiplication  of  Whole  Numbers  by  Fractions. 

III.  Ex.     Multiply  8  by  |. 
4  OPERATION.  8  multipHed  by  5  is  5  X  8  j   if  it  is 

0X5 4X5 az    J        multiplied  by  |,  a  number  one  sixth  as 

0  3         ^'         '    large  as  5,  the  product  must  be  one 

3  sixth  as  large  as  if  5  had  been  the  mul- 

tiplier, or  i  of  5  X  8.     The  expression  then  becomes  8  times  5,  di^ided 
by  6;  after  cancelling,—^  z=  —  =:  6f ,  Ans.     Hence  the 

o  o 

Rule.  To  multiply  a  whole  number  by  a  fraction  ;  — Multiply 
the  whole  number  by  the  numerator  of  the  fraction,  and  divide 
that  product  by  the  denominator. 

Examples. 


Multiply 

1.  36  by     |.    Ans.  24. 

2.  568  by    |.    Ans.  473^. 

3.  385  by  ^■^. 


4.  3681    by    ^V 

5.  5432    by    ^J. 

6.  87036    by    ^V 


7.  What  cost  I  of  1  ton  of  hay,  at  $12  a  ton-? 
Operation.  If  1  ton  of  hay  costs  $12,  ^  of  a  ton 

1.50  will  cost  I  of  $  12,  and  |  of  a  ton  will 

$3^^.00  X  5__  ^  ^ ^^  j^^^^     cost  5  times  |  of  $  12.    Cancelling,  we 
$  •     J         •     have  $1.50X5  =  $  7.50,  ^rw. 

8.  What  cost  f  of  an  acre  of  land,  at  $100  an  acre  ? 

9.  What  cost  y^2  of  an  acre  of  land,  at  $150  an  acre  ? 

10.  What  cost  f  of  a  gross  of  pens,  at  $.96  a  gross  ? 

11.  What  cost  5^  cords  of  wood,  at  $7.56  a  cord? 

12.  What  cost  3|  hogsheads  of  molasses,  at  $18.80  a  hogshead  ? 


MULTIPLICATION  OF  FRACTIONS.  73 

13.  What  cost  2f  firkins  of  butter,  at  $12.60  a  firkin? 

14.  "What  cost  63f  yards  of  flannel,  at  $.54  a  yard? 
1^  For  Dictation  Exercises>  see  Key. 

130*       MULTIPLICATIOJf   OP    FRACTIONS   BY    FRACTIONS. 

III.  Ex.     Multiply  |  by  |. 

Opehation,  i  multiplied  by  8,  is  8  times  | ;  if  it  be  multi- 

5  X  ^       20  plied  by  f ,  a  number  one  ninth  as  large  as  8,  the 

^  =  rr?  ^^*    product  must  be  one  ninth  as  large  as  if  8  had 

3  been  the  multiplier,  or  one  ninth  of  8  times  |. 

The  expression  then  becomes  ^^  j  after  cancelling,  g— -^  =.  ^^  Ans. 
Hence  the 

RcLfi,  To  multiply  a  fraction  by  a  fraction ;  —  Multiply  the 
numerators  together  for  a  new  numerator^  cmd  the  denominators 
for  a  new  denominator^ 

Examples. 
Mvdtiply 

4.  If  by      If. 

5.  If   by    l^f, 

6.  Ill   by-VsV-- 
7.     _8^  X  21=  what? 

Note.  — Reduce  mixed  numbers  to  improper  fractions  before  multiply- 
ing by  fractions. 

8.     18f  X   ^1  r=  ?        i         10.     15|  X     t\  =:  ? 


1. 

1  by 

f- 

Ans.  ^, 

2. 

f   by 

iV 

Ans.  f  J. 

3. 

IS  by 

?• 

Ans,  -^7f. 

9.      5i2Xjf7-  =  ?  I  11.        3|X31|=:r? 

Note,  —  In  the  ninth  example,  first  reduce  1^  to  lower  terms. 

12.     38§|X^f  =  ?         J         13.     2^X111^? 

14.  What  cost  2j^jy  boxes  of  raisins,  at  If  dollars  a  box  ? 

15.  What  cost  10 J  tons  of  coal,  at  $7|  a  ton? 

16.  What  cost  bh  pounds  of  coffee,  at  j^jj  of  a  dollar  a  pound  ? 

17.  What  cost   11-^^  pounds  of  pork,  at  ^^  of  a  dollar  a 
pound  ? 

^F"  For  Dictation  Exercises,  see  Key. 

131.     Reduction   of   Compound    Fractions  to   Simple 
Fractions. 
III.  Ex.,  I.     If  1  dollar  buys  f  of  a  yard  of  cloth,  how  much 
will  f  of  a  dollar  buy  ? 


74:  COMMON  FRACTIONS. 

Operation.  Explanation  1st.    If  one  dollar  buys  | 

3X3  __  9     J      ,         of  a  yard  of  cloth,  ^  of  a  dollar  will  buy  ^ 
6  X"4  "~  20  5   •»         *    of  I  of  a  yard,  and  f  of  a  dollar  Avill  buy  3 
times  ^  of  I  of  a  yard,  zrz  ^^^  of  a  yard. 

Explanation  2d.  If  one  dollar  buys  |  of  a  yard  of  cloth,  | 
of  a  dollar  will  buy  f  of  |  of  a  yard.  ^  of  ^  of  a  yard  is  ^g-  of  a  yard, 
which  may  be  shown  by  dividing  |  of  a  yard  into  4  equal  parts.  The 
whole  yard  can  thus  be  divided  into  4  X  5,  or  20  equal  parts  ;  there- 
fore, 1  part  will  be  -^^  of  the  whole  yard.  If  ^  of  ^  :=  ^L,  \  of  f  musf 
be  -^^j  and  f  of  |  must  be  3  times  ^3^,  or  -t^^.    Ans. 

III.  Ex.,  II.     f  of  -f  =  what?  L><J  —  1     Jus 

J  of  f  is  a  compound  fraction  (Art.  116),  and  is  reduced  to  a 
simple  fraction  bi/  multiplication.  Let  the  explanation  be  similar 
to  the  second  explanation  of  the  illustrative  example. 

Examples. 

1.  I  of -i-^  =r  what  ?  Ans.%, 

2.  f  of  f  of  ^\  =z  what  ?  Ans.  f 

3.  ^  of  §  of  f  of  I  of  f  =  what  ? 

4.  ^5  of  if  of  II  of  U  =  what? 

5.  7  of  2-1  of  5^  of  3^  =  what? 

6      ^  of  3f  of  l^f  of  60  =  what? 
^^  For  Dictation  Exercises,  see  Key. 

13d.     Division  of  Fractions  by  Whole  Numbers. 

As  dividing  the  numerator  of  a  fraction  diminishes  the  number 
of  parts,  their  size  remaining  the  same,  and  multiplying  the  de- 
nominator diminishes  the  size  of  the  parts,  their  number  remain- 
ing the  same  (Art.  119),  it  follows,  that 

To  divide  a  fraction  by  a  whole  number,  we  may  either  divide 
•*-  jumerator  hy  the  whole  number,  or  multiply  the  denominator. 

Examples. 
Divide 

ins.  \{. 


\. 

A  by  4- 

Ans. 

A- 

4. 

+^by  4 

2. 

f    by  3. 

Ans. 

f- 

5. 

fiby  3 

3. 

\%  by  7. 

Ans. 

k\- 

6. 

iJ  by  4 

DIVISION   OF   FRACTIONS.  76 


7.  ifii-f-       9=  what? 

8-     ^bWx^-^    11==  what? 
9.         4i2|  _i_  864  =  what  ? 


10.  .  A%  —    36  —  what? 

11.  15A^7    _!.      9  —  what? 

J^w5.  l^^f. 


Note.  —  In  example  11,  divide  15  by  9,  then  reduce  the  remainder  to 
an  improper  fraction,  and  divide  the  fraction  by  9. 

12.  Divide  28f  f  by  18.         |         13.     Divide  44if  by  368. 
^^  For  Dictation  Exercises,  see  Key. 

133.     Division   of  Whole  Numbers  and  Fractions  by 
Fractions. 

III.  Ex.,  I.     How  many  times  is  ^  contained  in  3  ? 

Solution.  ^  is  contained  in  any  number  twice  as  many  times  as  1 
is  contained  in  it.  1  is  contained  in  3,  3  times ;  .*.  ^  is  contained  in  3, 
2X3  =  6  times. 

Examples. 

1.  How  many  times  is  ^  contained  in  6  ?  Ans.  18. 

2.  How  many  times  is  j-  contained  in  18  ?  Ans.  126. 

3.  How  many  times  is  |  contained  in  8  ? 

How  do  you  divide  by  a  fraction  having  1  for  its  numerator  ? 
Divide 

4.  20  by    f        7.       56  by  yV-  1^-     1^1  by  gV- 

5.  27  by  j\.        8.     100  by  ^V  11.       96  by  ^V- 

6.  31  by  ^\,        9.     702  by  f  12.     108  by  ^V- 

13.  If  it  takes  J-  of  a  yard  of  cloth  to  make  a  vest,  how  many 
vests  can  be  made  from  8  yards  ? 

14.  If  a  man  walks  1  mile  in  ^  of  an  hour,  how  many  miles 
will  he  walk  in  8  hours  ? 

15.  If  the  cars  can  run  a  mile  in  ■r^^  of  an  hour,  how  many 
miles  can  they  run  in  10  hours  ? 

1 6.  At  1^  of  a  dollar  a  dozen,  how  many  dozen  eggs  can  be 
bought  for  10  dollars  ? 

III.  Ex.,  II.  Divide  4  by  f ;  that  is,  see  how  many  times  f 
is  contained  in  4. 

Operatiox.  .         . 
4  X  5       20                                     contamed  in  4,  4  times ;  ^  is  con- 
z=  —  =  6§,  Ans.   tained  in  4,  5  times  as  many  times  as  1 

"^  *J  is  contained  in  it,  or  5  X  4  times,  aaid  |, 


7&: 


COMMON  FRACTIONS. 


which  is  3  times  ^,  can  be  contained  in  it  only  ^  as  many  times  as  ^ 
is  contained  in  it,  or  ^  of  5  X  4  tunes. 

III.  Ex.,  III.     Divide  |  by  |. 

1  is  contained  in  f ,  f  of  a  time  ;  ^  is  contained 
in  |,  6  X  f  times  ;  |  (which  is  5  times  i)  is  con- 
tained in  it  -^  of  6  X  f  times.  The  expression  then 

4 


Operation. 
2 


^X*^  =  ±,Jns. 


0X5 


becomes  ^^    ;  after  cancelling,  AK^ ::::;^  1. ,  Ans. 

3  /\  5  5  5 

Hence  the 


Rule.  To  divide  a  whole  number  or  a  fraction  by  a  frac- 
tion ;  Multiply  the  dividend  hy  the  denominator  of  the  divisor, 
and  divide  by  the  numerator  ;  or, 

Invert  the  divisor,  and  proceed  as  in  Multiplication. 


Examples. 


17. 
18. 
19. 
20. 
21. 
22. 
23. 


18 


21 -^A  =  ? 


f  =  ?  Ans.  21  f. 


Ans.  70. 
Ans.  140. 
Ans.  96. 
108~-if  =  ?    ^W5. 153. 


98 


54~-3%  =  ? 


if-:? 


31.    2|- 


24. 
25. 
26. 
27. 
28. 
29. 
30. 

If  r=? 
$X  1 


]  06 


tV  =  ? 


■nix 

18 
TUT 


OrEBATiON.   2^-M?=:f-T-Y-  =  ^-^^^  =  }=lf,  ^te. 

2 
Note.  —  Reduce  mixed  numbers  to  improper  fractions  before  dividing. 

32.  3f  -^  4^-2^  =  ?  34.  26^  -f-  3^^-  =  ? 

33.  5j^-^6^=:?  35.  1  -^541f  =? 

36.  fof|^fof^V=? 
Operation. 

3X8^2X  7  ^g{X^Xg(x/^_       . 
5  X  9  •  3  X  15      ^X^X^X  7  -*^'^^^' 

Ans.  45j^. 


|of|-r-5ofA 


87.*  f  of  1-^  ~  A  of  f  of  I  ? 

38*  fr  of  A  of  n  -T-  /j  of  if  of  /-s  ? 


COMPLEX  FRACTIONS. 


77 


«^r  71§  X  A  -^  f  of  7^?  ^n*.  12^%. 

40.*  How  many  sevenths  are  there  in  ^  of  7^^  -r  6§? 
41*  llow  many  times  ^  of  14  in  2^%-  X  f  of  81  ? 
I^^  For  Dictation  Exercises,  see  Key. 

134.   Reduction   op   Complex  Fractions   to  Simple 
Fractions. 

Ill    Ex.     —  =:  what?       This  is  a  complex  fraction  (Art.  117), 
2"^  and  is  reduced  hi/ performing  the  divU 

sion  indicated;  thus,  3 


1.  —  :=what? 


2.  |-=:  what? 

3.  -^  =  what? 
2f 


Examples 
Ans.  6| 

Ans.  yV 

Ans.  2 


'TU* 


4. 


86/^ 


what  ? 


5tV 

8-9- 
9.*  13   X    '' 


5.  ^~  what? 

TT 

6.  ^??i==what? 
7683^ 

74  X  6  X  tV 


8: 


xVof  3f 


what  ? 


6? 


Q    XxT-  X  707=  what? 

lOr  If  ^^/g^^^    is  the  dividend,  and  gg  ^  j^  ^  ^g  the 

divisor,  what  is  the  quotient  ? 

lit  If  1-  is  the  divisor,  and  4i  the  quotient,  what 

^T  X  ^g-  X  /t  9  9 
is  the  dividend  ?  ^"~7^ 

12!  If  ^^^  X  5^  X  4  .^  ^^^^  dividend,  and  18|  the  quotient, 
2t  X  t\  X  l^G 
what  is  the  divisor? 

J^*  For  Dictation  Exercises,  see  Key. 


78  COMMON  FRACTIONS. 

ISS*    To  FIND  THE  Whole  Number  when  a  Fractional 
Part  of  it  is  given. 

III.  Ex.     38§  is  ^  of  what  number  ? 

Operation.  38  2  — -  iii .  jf  115 

„„,       116      X%X5       145       ,^,     ,  ki  of  some  number, 

38f  =  -i7-.    —^ — — — ,  = —r- =  48^,  ^W5.     -^  of  that  number  IS 

'^  ^     X  ^  ^  i  of  H^,  and  I,  or 

the  entire  number,  is  5  times  ^  of  -^-^  j  cancelling,  the  expression  be- 

29X5        ^01      A 
comes  — — -  =  48|,  Ans. 

Examples. 

1.  16^  is  -^xT  of  what  number?  Ans.  18. 

2.  25^  is  §  of  what  number?  Ans.  38|. 
^'    F  of  ^1  is  f  of  what  number?  Ans.  l^^* 

4.  f  of  6f  is  f  of  what  number?  Ans.  13^. 

5.  y\  of  ^Tj-  is  f  of  what  number  ? 

6.  2^  X  7|  is  3^  times  what  number  (or  J-  of  what  number)?, 

7.  182  -^  12  X  H  is  3^  times  what  number? 

8.  From  New  York  to  Troy  is  150  miles,  which  is  f  of  the 
distance  from  New  York  to  Rouse's  Point ;  what  is  the  distance 
from  New  York  to  Rouse's  Point  ?  Ans.  350  miles. 

9.  Mr.  Aborn  owns  f  ^  of  an  acre  of  land ;  his  neighbor  Jones 
owns  f  as  much,  which  is  ^  of  what  Mr.  Green  owns ;  what  does 
Mr.  Green  own  ? 

10.  If  f  of  a  piece  of  work  be  performed  in  25  dajs,  what 
number  of  days  will  be  required  to  do  the  remainder  ? 

11.  Paid  $6  a  week  for  board  in  Boston,  which  was  f  of  what 
I  paid  in  New  York ;  this  was  ^  of  what  I  paid  in  Philadelphia ; 
and  this  was  f  of  what  I  paid  in  Washington.  What  did  I  pay 
in  Washington  ? 

12.  A  vessel  having  lost  ^  of  her  cable,  has  200  feet  remain- 
ing ;  how  many  feet  had  she  at  first  ? 

Solution.  —  If  \  be  lost,  |  will  remain  ;  if  ^  =  200  feet,  i  =  i  of 
200  feet,  and  |,  or  the  whole  cable,  =  5  X  i  of  200  feet,  =  250  feet. 

13.  A  ship's  crew  having  lost  ^  of  their  bread,  are  obliged  to 
subsist  on  14  ounces  a  day;  what  were  they  allowed  at  first? 


EXAMPLES.  79 

14  Having  lost  tt  ^f  my  money  in  trade,  I  now  have 
42476.50  ;  what  had  I  at  first? 

15.  A  mother  and  her  son  together  have  $45  in  a  purse ;  the 
son's  part  is  f  as  great  as  the  mother's.  Required  the  part  of 
each  ? 

SoLUTiox. ; — The  purse  contained  once  the  mother's  money  and  f  as 
much  more,  (the  son's).  If  1|  (|)  times  the  mother's  part  =  $45, 
^  z=  ^  of  $45,  and  f  =  3  X  i  of  $45,  zzz  $27,  the  mother's  part ; 
I  =z  2  X  1  of  $45,  =1  $18,  the  son's  part. 

16.  The  sum  of  the  ages  of  a  father  and  son  is  155  years,  the 
son's  age  being  ^  the  age  of  the  father ;  what  is  the  age  of  each  ? 

17.  A  body  of  4800  troops  has  ^  as  many  cavalry  as  infantry ; 
what  is  the  number  of  each  ? 

18.  A  lot  of  land  yielded  4140  bushels  of  grain  in  two  years, 
yielding  |  as  much  the  second  year  as  the  first ;  what  was  the 
yield  each  year? 

19.  What  number  is  that  to  which  if  f  of  itself  be  added  the 
sum  will  equal  275  ? 

20.  A  carpenter,  who  has  a  number  of  floors  to  lay,  estimates 
that  it  will  cost  §  more  to  lay  them  with  hard  pine,  worth  $28  a 
thousand  feet,  than  with  white  pine ;  what  is  the  price  of  white 
pine  per  thousand  ? 

21.  In  counting  his  fowls,  a  farmer  finds  that  he  has  396  in 
all,  which  is  ^  more  than  he  had  the  previous  year ;  how  many 
had  he  then  ? 

22.  He  has  sold  his  eggs  at  an  average  of  13j^  cents  per 
dozen,  which  is  ^  higher  than  the  previous  year ;  what  did  they 
average  then  ? 

23.  He  is  paid  for  grain  $1|  per  bag,  which  is  ^  less  than  ho 
was  paid  last  year ;  what  was  he  paid  last  year  ? 

24*  Mr.  Ober  owns  504  acres  of  wood  land,  |  of  which  he 
exchanges  with  Mr.  Fay  for  iO^  acres  of  meadow  land,  which  is 
|-  of  what  Mr.  Fay  owned  ;  how  much  meadow  land  did  Mr.  Fay 
have  at  first?  How  much  wood  land  after  the  exchange  was  made  ? 

i^  For  Dictation  Exercises,  see  Key, 


80 


COMMON  FRACTIONS. 


136*       To    FIND   WHAT   PaRT    ONE   NuMBER    IS    OF 
ANOTHER. 
Note.  ^  Yoxinger  pupils  may  omit  this  article. 

What  part  of  2  is  1,  or  1  is  what  part  of  2  ?     Ans.  1  is  J  of  2; 
because  it  is  1  of  the  2  equal  parts  into  which  2  may  be  divided. 
1  is  what  part  of  3  ?  of  5  ?  of  7  ?  of  9  ?  of  8  ?  why  ? 
1  is  what  part  of  19  ?  of  11  ?  of  6  ?  of  15  ?  of  33  ? 

iLLrSTKATIVE   EXAMPLES. 

1.  3  is  what  part  of  10  ?    1  is  J^  of  10, .-.  3  must  be  y%of  10.    Ans.  j%, 

2.  I  is  what  part  of  7  ? 

^  OPERATioji.  1  is  I  of  7 J  .-.  I  of  1  must  be  I  of  I  of  7,  or 

7-^3  =  A' ^"*-    -j'rof^. 

3.  What  part  off  is  1? 

Operation.    |^  is  ^  of  |,  .-.  f  or  1  whole  one  is  |  of  f .  Ans,  |. 

4.  What  part  of  |  is  2  ? 
I  is  I  of  I,  .'.  I,  or  1  is  |  of  |,  and  2  must 

Ans.  i^. 


Opekatiox. 

2X9 


=  VS  ^^^'      be  2  X  f  r  or  Jf  ■  of  |. 
5.  What  part  of  i^is|? 

iV  is  t\  of  1^,  .-.  If,  or  1  is  If  of  1|,  and  | 
5^1,  Am.    iBust  be  |  of  f  f  of  i|,  or  ||  of  i^.  ^W5.  ||. 

From  the  above  we  derive  the  following 

Rule.     To  ascertain  what  part  one  number  is  of  another: 
Divide  the  mimber  expressing  the  part,  by  that  of  which  it  is  a  part. 

Examples. 


Opekatiox. 

liXi  — 51 

11X5  "' 


What  part  of 

1.      5  is  3?     .^lis.  f. 

i   7. 

2.       8  is  6? 

8. 

3.     10  is  7  ? 

9. 

4.     20  is  15? 

10. 

5.     30  is  75? 

11; 

6.     89  is  267? 

12. 

2|  is  1  ?  . 

J^fisif? 
3|is2|? 


8 is  I?  Ans. -^^  IS. 
Ilis||?^w5.^.  14. 
48  is    5|?  '     15. 

19isl2|?  16. 

|isl?  Ans.^.  17. 
l\isl? 

18.  If  by  a  pipe  a  cistern  can  be  filled  in  3  hours,,  what  part  of  the 
cistern  will  be  filled  in  1  hour  ?  in  2  hours  ? 

19.  If  a  piece  of  work  can  be  performed  in  9  days,  what  part  of  the 
work  can  be  performed  in  1  days? 

20.  A  can  perform  a  journey  on  foot  in  7-J  days  j  what  part  of  it  can 
he  perform  in  2 J  days  ? 


MULTIPLES.  81 

21.  Mr.  Bailey  has  $54,  and  pays  $18  for  a  coat ;  what  part  of  his 
money  does  he  spend  ? 

22.  Charles  picks  2|  quarts  of  blackberries,  and  Eben  5f  quarts. 
If  Eben's  blackberries  are  worth  one  dollar,  what  part  of  a  dollar  are 
Charles's  worth? 

23.  A  and  B  hired  a  pasture  together.  A  pastured  12  cows,  and  B 
13  cows  in  it ;  what  part  of  the  price  should  each  pay  ? 

24.  Four  men  were  hired  to  work  on  a  farm ;  A  mowed  7  acres  ;  B 
mowed  5  acres  ;  C,  4  acres,  and  D,  2  acres.  They  received  $27.  What 
was  each  one's  share  ? 

25.  Mr.  Snow,  dying,  left  $75000  to  his  wife  and  three  sons.  To  his 
wife,  $30,000  ;  to  his  oldest  son  just  as  large  a  part  of  the  remainder 
as  his  wife's  portion  was  of  the  entire  property;  to  his  2d  son  |-  of 
w^hat  his  eldest  received,  and  to  his  youngest  the  rest.  What  was  eacb 
son's  share  ? 

1^="  For  Dictation  Exercises,  see  Key. 

Multiples  op  Numbers. 

137.  A  Multiple  of  a  number  is  any  number  that  will  con- 
tain it  without  a  remainder;  thus,  8,  12,  16,  and  20,  are  multi- 
ples of  4. 

138.  A  Common  Multiple  of  two  or  more  numbers  is  a 
number  that  will  contain  each  of  them  without  a  remainder; 
thus,  20  is  a  common  multiple  of  5  and  2. 

lS9o  The  Least  Common  Multiple  of  two  or  more  num- 
bers is  the  least  number  that  will  contain  each  of  them  without  a 
remainder ;  thus,  10  is  the  L.  C.  M.*  of  2  and  5. 

Exercises. 

Name  any  6  multiples  of  5.  Name  3  multiples  of  12.  Name  all 
the  multiples  of  11  up  to  140.  Name  any  common  multiple  of  10 
and  6.     Of  3,  6,  and  5. 

14:0«     To   FIND   THE    Least  Common  Multiple  of  two 
OR  MORE  Numbers. 

Tha  common  multiple  of  two  or  more  numbers  must  contaia 
*  Least  Common  Multiple. 


82  COIOION  FRACTIONS. 

all  the  factors  of  those  numbers,  and  the  least  number  that  con« 
tains  all  their  factors  must  be  the  least  common  multiple, 

III.  Ex.     Find  the  L.  C.  M.  of  4,  6,  10  and  15. 
Operation.  We  find  the  factors  of 

4  =  2X2  4  to  be  2  and  2,  of  6  tto 

6  =  2X3  be  2  and  3,  of  10  to  be  2 

10  =  2  X  5  and  5,  of  \o  to  be  3  and  5. 

15  =  3  X  5  To   contain   4,  the  L.  C. 

L.  C.  M  =  2  X  2  X  3  X  5  =r  60,  Ans.  M.  must  contain  the  fac- 
tors 2  and  2,  which  we  note.  To  contain  6,  it  must  contain  2  and  3  ; 
we  have  already  noted  2,  so  we  need  introduce  only  ihe  3.  To  contain 
10,  it  must  contain  2  and  o  ;  as  we  have  noted  2,  we  introduce  only 
the  5.  To  contain  15,  it  must  contain  3  and  5  ;  we  have  already  noted 
these  fiictors,  .-.  2  X  2  X  3  X  5  =  60,  must  be  the  L.  C.  M.    Hence  the 

Rule.  To  find  the  L.  C.  M.  of  two  or  more  numbers: 
Separate  the  numbers  into  their  prime  factors.  Find  the  product 
of  all  the  different  prime  factors^  taking  each  factor  the  greatest 
number  of  times  it  is  contained  in  any  one  number. 

Examples. 
Find  the  L.  C.  M.  of 

1.  8,  18,  20,  and  21.  Ans.  2520. 

2.  12,  16,  and  28.  Ans.  336. 

3.  3,  5,  8,  12,  20,  36,  and  45.  Ans.  360. 
Note.  —  "When  one  of  the  given  numbers  is  contained  in  another,  it  may 

be  disregarded  in  the  operation ;  thus,  in  the  preceding  example,  3,  5,  and 
12  may  be  rejected.     Why  ? 


Find  the  L.  C.  M.  of 

4.  18,  36,  40,  60,  and  72. 

5.  12,  16,  42,  56,  and  70. 

6.  13,  28,  35,  39,  and  49. 


7.  9,  18,  32,  48,  and  52. 

8.  8,  16,  28,  35,  and  63. 

9.  Of  the  nine  digits. 


When  several  numbers  are  prime  to  each  other,  what  must 
their  L.  C.  M.  equal  ? 

14:1*  The  above  is  the  better  method  for  finding  the  L.  C.  M. 
when  the  numbers  are  easily  separated  into  their  prime  factors. 
For  larger  and  more  difficult  numbers  observie  the  following 
method  :  — 


MULTIPLES. 


83 


III.  Ex.     Find  the  L.  C.  M.  of  3G,  112,  76,  and  60. 

Here,   by 


Opeuatiox. 
2  )  36^12,  76,  60 

2  )i87~56,  38,_30 

3  )  9,    28,l9,J5 

3,    28,  19,"5 
L.  C.  M.r=2  X  2  X  3  X  3  X  28  X  19  X  5  =95760. 


re- 
peated divisions, 
we  take  out  all 
the  factors  that 
are  common,  2, 
2,  and  3;  the 
least      common 


multiple  must  contain  these  factors  and  those  which  aife  not  common  j 
.-.  2  X  2  X  3  X  3  X  28  X  19  X  5::i=95760,  must  be  the  L.  C.  M. 
souglft.     Hence  the 

Rule.  To  find  the  L.  C  M.  of  two  or  more  numbers:  Divide 
hy  any  prime  factor  which  is  contained  in  two  or  more  of  the 
numbers  without  a  remainder,  writing  the  quotient  and  undivided 
numbers  in  a  line  beneath,  and  thus  proceed  till  no  two  numbers 
can  be  divided  by  the  same  prime.  The  product  of  all  the  divisors 
and  the  numbers  remaining  is  the  L,  G.  M. 

Examples. 


Find  the  L.  C.  M.  of  the  following: 


13!  2784,  147,  and  472. 
14t  912,  9500,  and  855. 
15!  1146,  1936,  and  24. 
16!  880,  9680,  and  8624. 
17r  539,  573,  and  9680. 


10.  338,  364,  and  448. 

Ans,  75  712. 

11.  184,  390,  and  552. 

Ans.  35,880. 

12.  847,  968,  and  1001. 

Ans,  88,088. 

18.  What  is  the  width  of  the  narrowest  street,  across  which 
stepping  stones  either  4,  5,  or  8  feet  long  will  exactly  reach  ? 

19.  What  is  the  narrowest  box  that  will  exactly  pack  ribbons 
either  3,  4,  or  5  inches  wide  ? 

20.  What  is  the  smallest  bill  that  may  be  paid  by  using  either 
dimes,  three-cent  pieces,  or  quarter  dollars  ? 

21.  What  is  the  smallest-sized  cistern  the  contents  of  which 
may  be  exactly  measured  by  using  either  15,  28,  or  36  gallon 
casks  ? 

1^  For  P|ct^tipi>  JExercises,  see  Key, 


84  COMMON  FRACTIONS. 

143(     Eeduction  of  Fractions  to  Equivalent  Feao 
TiONs  having  a  Common  Denominator. 

"When  the  denominators  of  fractions  are  alike,  they  are  said  to 
have  a  Common  Denominator. 

III.  Ex.  Reduce  |,  |,  and  ^  to  fractions  having  a  common 
denominator. 

"We  can  change  these  fractions  to  fractions  of  any  given  denom- 
inator ;  but  tht  most  convenient  denominator  for  most  purposes  is 
that  which  is  the  least  common  multiple  of  the  denominators  of  tli% 
given  fractions ;  and,  in  the  following  examples,  such  denoraiflators 
are  always  required.  In  the  preceding  example,  we  must  first  find  the 
L.  C.  M.  of  3,  4,  and  6,  whjch  is  12 ;  and  then  reduce  f ,  |,  and  |  to 
twelfths.  1=1|,  .-.  |=r^  of  If  or -j^,  and  |  =  ii/^^z=-jV  By 
the  same  process  we  fijid  that  f  r=  3^,  and  |  =  \^.  Ans.  -^,  -^-^^  i^. 
Hence  the 

Rule.  To  reduce  fractions  to  equivalent  fractions  having  a 
common  denominator :  Reduce  the  fractions,  to  their  simplest 
forms ;  find  the  least  common  multiple  of  the  denominators  for 
the  common  denominator  ;  multiply  the  numerator  of  each  fraction 
hy  the  number  hy  which  you  would  multiply  its  denominator  to 
produce  the  common  denominator,"^  The  respective  products  will 
he  the  numerators  of  the  required  fractions, 

III.  Ex.     Reduce  ^,  -j^l,  and  f  to  fractions  having  the  L.  C.  D. 

Entire  Opkration. 
8  =  2X2X2  i^^xTEJ^ii. 

12^:2X2X3  -i-V  =  ^-W-^--  f  f  • 

9=3X3  f  =  2  x_2^x^2.x 2  —  |6. 

L.  C.  M.=:2  X2X2X3X3=:72. 

Here  72  is  the  L.  C.  M. ;  and  as  8  =  2  X  2  X  2,  it  must  be  multi- 
plied by  3  X  3  to  produce  72,  .-.  \  =  J^^^,  and  |  =  .&i<^I  =  4|. 

Show  why  yV  =  H  ;  why  |  z=  ||. 

*  If  that  number  is  not  readily  seen,  it  may  be  found  by  dividing  the 
toramon  denominator  by  the  denominatpr  of  the  priginal  fraction. 


ADDITION  OF  FRACTIONS. 


85 


Examples. 
Reduce  the  following  to  fractions  having  the  least  common 
denominator:  — 


4.  ^\,  i§,  and  A-t 

5-  T5r  ih  1^5-'  and  sV 

6-  /(J,  tVi  -3%^  and  ^2-. 


7        3  5         2  8       nnrl      3  2 


1.  J-,  f ,  and  f. 

2.  f  Z^-.  and  /t- 

3.  t'ttj  2^,  and  -j^^. 
1^="  For  Dictation  Exercises,  see  Key.' 

143.     Addition   of   Fractions 
Examples. 


X 


f  +  i^what? 
f-j-f  zrivvhat? 
f  +  f  =what? 


Ans,  f. 


4.  36^ -f  ^ij- :=  what  ? 


T^or 


+  t5(J  =  what  ? 


6.  A+ini  — what? 


The  above  examples  are  easily  performed,  as  the  quantities  to 
be  operated  upon  are  like  quantities,  that  is,  have  the  same  de- 
nominator. In  such  cases,  we  have  only  to  add  the  numerators. 
When  fractions  of  different  denominators  are  to  be  added,  thii/ 
must  first  be  reduced  to  fractions  having  a  common  denominator^ 

III.  Ex.  —  Add  J  and  ■^^. 

£ 16       £ 15  lO-i-15 n 

9         36*       12        36*  36  36* 


Operation.    ~z=z  —      —z=z—. 


Ans. 


7.  -1  +  ^  +  1  =  ? 
9-  f+A+i  +  l 

10.  f  +  l 4-^  +  ^8=? 

11.  i*5  +  A  +  i  +  l=? 

12./.  +  l  +  i=? 
13.  A  +  /a  +  A  =  ? 


•ItV 
Am.  2^. 


14-  A  +  A  +  i!=? 

15.  «  +  A  +  tV  =  ? 

16.  f  +  S  +  4§=? 

17.  A  +  a^  +  A  =  ? 

18.  A+g\+A  +  ii=? 

19*t'6  +  A  +  3'2+/T  =  ? 


Note.  —  Add  the  whole  numbers  and  fractions  of  the  following,  and 
similar  examples,  separately. 


21.36+21  +  7,^2=? 
22.  18§  +  16^4-28f  =  ? 


23.  272^+16^  +  181  =  ? 
24!  104^3^  + 8/0 +  480|i  =  ? 


t  What  operation  should  j&rst  be  performed  upon  these  fractions  ? 


8(5 


COMMON  FRACTIONS. 


25.  2007y3(y  +  1070|  +  8040|  =  ? 
27.%vof2|  +  iof|  +  f  of2^V  = 


28r2/^  +  ^of/3  +  ^  =  ? 
29MA  +  lT\+i|=what? 


Ans.  7^V% 


For  Dictation  Exercises,  see  Key. 

144.     Subtraction   of   Fractions. 
Examples. 


1. 

f— 2-?                        ^,?«.  f 

5.    TV^i-Tli  =  ? 

2. 

i-|  =  ?       Ans.^  =  i. 

6.  t¥^-tIf  =  ? 

3. 

j'^-j\  =  ? 

7.^-1  =  ? 

4. 

T^TT T§(T  =  ? 

Ans.  ii. 


Note. — The  denominators  in  Example  7  being  unlike,  the  fractions 
must  be  reduced  to  fractions  having  the  same  denominator. 

8.  f-f  =  ? 


9.  ^- 

10.  A 

11.  1-^^-? 


TT 

3 

"5 


—■7 


Ans.  ^. 
Ans.  ^V- 


T3- 


12. 

A- 

-A 

—  ? 

13. 

A- 

-T^j: 

=  ? 

14. 

2§- 

-li  = 

=  ? 

Note. 
fractions. 


Subtract  without  changing  the  mixed  numbers  to  improper 


18.  18^  —  15^3  =  ? 


19.    17^— 12^9^  =  ?. ^715.43. 


15.  8-L  — 3f  =  ?       Ans.  6^. 

16.  7|  — 2J^=r?    Ans.b:^. 

17.  103_5^|=:? 
Note.  — As  -^-^  cannot  be  taken  from  ■^,  it  will  be  necessary  to  reduce 

1  of  the  17  to  Iialves,  making  the  minuend  16|,  when  subtraction  can  be 
lasily  performed. 


20.  2^  — 1|  =  ? 


2f 


Ans.  ^. 
?  Ans.  14j-|. 


21.  17i 

22.  12^  — f  r=? 

23.  26f— If  =  ? 

24.  19  — 2|z=?      Ans.  16|. 


25.  36  —  1=:? 

26.  75  — lo^zzr? 

27.  l-^of^^ 


•g  —  ? 
16 


28.  18f-^f-i|of2^=i:what? 


For  Dictation  Exercises,  see  Key. 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS.  87 

14^.    Addition   and    Subtraction   of    Fractions 
Combined. 

Give  a  rule  for  the  addition  of  fractions  ;  for  subtraction. 


Examples. 

'' 

1. 

I  +  A-A 

=  ? 

^715 

'■  m- 

2. 

i- 

-1+1  = 

V 

3. 

1- 

-i  +  i- 

■i+i- 

-i  +  i- 

i+i- 

-tV 

Ans. 

mi- 

4. 

f- 

■i+ii- 

-Ti^  = 

:  what  ? 

5. 

1- 

■i-i- 

■i-T\ 

— ^v-^v 

-tJ.: 

—  ? 

6. 

1- 

■i-A- 

"rJff  — 

what? 

7. 

20- 

-S^  +  t' 

rof§  = 

-p 

8. 

8/t 

-2|  +  ' 

H=? 

9.  Ul  X  f  off +  10f— rVof  M  =?  ^^,.  18^^.1.. 

10.  7-(/^-^9^)  =  ?  ^  J,z..  Gif. 

11.  5-(t  +  3V)  =  ?  Ans.4iU. 

12.  A  man  receives  4^  per  cent,  commission  for  selling 
goods ;  he  pays  |  per  cent,  for  storage ;  what  per  cent,  does  lie 
retain  ? 

13.  If  he  receives  6f  per  cent,  for  selling  goods,  and  If  per 
cent,  for  insuring  their  sale,  and  pays  If  per  cent,  for  storage,  and 
y^2-  per  cent,  for  auctioneering ;  what  per  cent,  does  he  retain  ? 

14.  How  much  will  be  left  of  a  piece  of  cloth  containing  7 
yards,  after  cutting  from  it  2  vests  and  a  coat,  allowing  f  of  a 
yard  for  a  vest  and  4|^  yards  for  a  coat  ? 

15.  Bought  of  Mrs.  Frye  1  bonnet  for  $4.37^,  2  hats  at 
$2.12^  apiece,  4  yards  ribbon  at  $.16f  per  yard,  2  yards  ribbon 
at  33^  cents  a  yard,  and  gave  in  payment  a  ten  dollar  bill ;  what 
should  she  give  me  in  return  ? 

16.  From  8  apple  trees  I  gathered  as  follows:  2^  barrels,  5^ 
barrels,  5-|  barrels,  4|-  barrels,  3f  barrels,  If  barrels,  3^  barrels, 
and  2^  barrels.  I  sold  15  J-  barrels  to  one  man,  and  2^  barrels  to 
another,  how  many  barrels  had  I  left  ? 


88  COMMON  FRACTIONS. 

17*  To  what  must  you  add  the  difference  between  8f  and 
36/^,  that  the  amount  may  be  50§  ? 

18f  If  7|  X  f  —  '^Tua  is  the  minuend,  and  ^^^  the  remamder, 
what  is  the  subtrahend  ? 

1416.    Greatest  Common  Divisor  of  Fractions.  * 
III.  Ex.     Find  the  greatest  common  divisor  of  f ,  |,  and  |. 

OPEPwVTION. 

G.  C.  D.  of  6,  8,  and  4  z=  _2_    ^^^    -      We  find  the  G.  C.  D.  of  the 
L.  C.  M.  of  7,  9,  and  5  =  S15'  '      numerators  6,  8,  and  4  to  be 

2.  2  is  a  divisor  of  6,  but  must  be  divided  by  7  to  be  a  divisor  of  |. 
It  must  also  be  divided  by  9  to  be  a  divisor  of  f ,  and  by  5  to  be  a 
divisor  of  ^.  To  be  at  the  same  time  a  divisor  of  these  fractions,  it 
must  therefore  be  divided  by  7,  and  9,  and  5,  or  by  their  least  common 
■^lultiple.     Hence  the 

Rule.  To  find  the  G.  C.  D.  o^  f»*actions:  Reduce  the  frac- 
tions to  their  lowest  terms  ;  then  divide  the  G.  C.  D,  of  the  numer' 
cLtors  by  the  L.  C,  M.  of  the  denominators. 

EXAMPLI^.S. 

1.  Find  the  G.  C.  D.  of  §,  |,  and  |.  Ans.  yV- 

2.  Find  the  G.  C.  D.  of  y^^,  -r\,  and  2^  or  f .  Ans.  ^V 

3.  Find  the  G.  C.  D.  of  3^,  ^^,  |,  and  J-f." 

4.  Find  the  G.  C.  D.  of  f ,  f ,  and  4. 
Note.  —  4  can  be  regarded  as  i. 

5.  Find  the  G.  C.  D-  of  f  J,  ^3^,  f,  and  2. 

G.  What  is  the  size  of  the  largest  cup  which  is  an  exact  meas- 
ure of  1^,  1|,  8^,  and  ^  pints  ? 

7.  What  is  the  width  of  the  widest  carpeting  that  will  fit  4 
rooms  of  the  following  widths:  13^  feet,  21  feet,  31^  iaeX,  3G2 
feet  ? 

^^  For  Dictation  Exercises,  see  Key. 

147.    Least  Common  Multiple  of  Fractions.* 
III.  Ex.     Find  the  least  common  multiple  of  J-,  f,  and  f . 

^       __^  We  find  the""  L.  C.  M.  of  the 

Operation. 

L.  C.  M.  of  1,  3,  and  5  =  15  numerators,  1,  3,  and  5,  to  be 

G.  C.  D.  of  2,  4,  and  6  z=.T'  ^''^-      ^^'     ^^^  ^'^  ^^  "<^*  ^^'^^^  *« 

ascertain  the  least  number  that 

*  Articles  146  and  147  can  be  omitted  by  younger  pupils. 


QUESTIONS  FOR  REVIEW.  89 

will  contain  1,  3,  i  nd  5,  but  one  that  will  contain  ^,  |,  and  |.  To 
cont  iin  each  of  these  fractions  separately,  it  might  be  divided  by  2,  by 
4,  or  by  6 ;  but  to  contain  them  at  once,  it  can  be  divided  only  by 
theii-  G.  C.  D.     Hence  the 

EuLE.  To  find  the  L.  C.  M.  of  fractions :  Reduce  the  frac- 
tions to  their  lowest  terms,  then  divide  the  L.  C.  M.  of  the  numer- 
ators  hy  the  G.  C.  D.  of  the  denominators. 

Examples. 

1.  Find  the  L.  C.  M.  of  -^,  JL3^  and  7^.  Ans,  47 6f. 

2.  Find  the  L.  C.  M.  of  if,  ^  of  3^,  and  6.  Ans.  390 

3.  What  is  the  width  of  the  narrowest  cloth  that  can  be  cu* 
into  strips  either  f ,  IJ^,  or  4  inches  wide  ? 

4.  What  will  be  the  length  of  the  shortest  court  that  can  b< 
paved  with  stones  of  either  of  the  following  lengths,  viz.,  1^  fl.r 
2  ft.,  4  ft.,  or  2§  ft.  ?  Ans,  24  it. 

5.  What  must  be  the  width  of  the  narrowest  court  that  will 
receive  either  of  the  same  stones  widthwise,  their  widths  being 
1  ft.,  1^  ft.,  3  fl.,  and  2  ft.  ? 

6.  On  a  stringed  instrument  in  perfect  tune,  while  C  makes 
1  vibration,  D  makes  f ,  E  J,  F  |,  G  f ,  A  ^,  B  VS  and  C  2.  If 
all  are  struck  at  once,  in  how  many  vibrations  of  C  will  they  all 
again  coincide  ? 

7.  In  how  many  vibrations  of  C  will  C,  E,  G,  and  C  coincide  ? 
wUl  C  and  D  coincide  ?  C  and  E  ?  B  and  C  ?  C  and  O  ? 

^^  For  Dictation  Exercises,  see  Key. 

Questions  for  Review. 
Definitions  and  Properties  of  Numbers. — AVhat  is  the  sign 
for  plus  ?  for  minus  ?  for  greater  than  ?  less  than  ?  equal  to  ?  multi- 
plied by?  divided  by?  therefore?  What  does  a  parenthesis  or 
vinculum  signify  ?  What  are  integral  numbers  ?  What  are  frac- 
tional numbers  ?  mixed  numbers  ?  What  is  a  prime  number  ?  a  com- 
posite number  ?  What  are  the  factors  of  a  number  ?  What  is  a  prime 
factor  ?  A  composite  number  equals  what  product  ?  When  are  num- 
bers prime  to  each  other  ?  What  is  a  power  of  a  number  ?  What  is 
the  square  or  second  power  of  a  number  ?  the  fifth  power  ?  What  is 
a  root  of  a  number  ?  the  square  root  ?  the  cube  root  ?  the  sixth  root  ? 


90  COMMON  FRACTIONS. 

MTiat  i  s  the  sign  for  a  power  ?  for  a  root  ?  What  indicates  the  degree 
of  root  ?     What  is  an  even  number  ?  an  odd  ? 

Divisibility  of  Numbers.  —  When  are  numbers  divisible  by  2  ? 
by  3P  by  4P  by  5P  by  6?  by  8?  by  9?  by  10?  by  11?  by  any  com- 
posite number  ?  How  shall  we  ascertain  whether  any  given  number  is 
prime  ?     Describe  Eratosthenes'  sieve  ? 

Factoring  of  Numbers.  —  What  is  the  simplest  way  of  resolving 
numbers  into  their  prime  factors  ?  What  other  method  can  you  de- 
scribe, and  when  would  you  use  it  ?  Find  the  factors  of  180  by  first 
method,  and  explain  the  process.  Find  the  factors  of  10296  by  sec- 
ond method,  and  explain  the  process. 

Greatest  Common  Divisor. — What  is  a  divisor  of  a  number  ?  a  com. 
mon  divisor  of  two  or  more  numbers  ?  the  greatest  common  divisor  ? 
Find  the  G.  C.  D.  of  three  numbers  by  the  first  method  given.  Explain 
and  give  the  rule.  Find  the  G.  C.  D.  of  three  numbers  by  second 
method.  Explain  and  give  the  rule.  In  what  cases  is  the  second  method 
the  better  ?     When  is  it  necessary  to  find  the  G.  C.  D.  of  numbers  ? 

Fractions.  —  What  is  a  fraction  ?  Name  and  describe  its  terras. 
Name  the  different  kinds  of  fractions  of  which  you  have  learned. 
Define  a  common  fraction ;  a  decimal  fraction ;  a  proper  fraction  ;  an 
improper  fraction ;  a  mixed  number  ;  a  compound  fraction  ;  a  complej? 
fraction.  Give  an  example  of  each.  Explain  the  expression  ^.  Upon 
what  does  the  value  of  a  fraction  depend  ?  Which  of  the  fundamental 
rules  is  indicated  by  a  fraction  ?  What  effect  does  multiplying  the 
numerator  of  a  fraction  have  upon  that  fraction  ?  Why  ?  In  what 
other  way  could  you  produce  the  same  effect,  and  why  ?  What  effect 
does  dividing  the  numerator  have  upon  a  fraction  ?  Why  ?  In  what 
other  way  could  you  produce  the  same  effect,  and  why  ?  What  effect 
does  multiplying  both  terms  of  a  fraction  by  the  same  ntimber  have 
upon  it  ?  AVhy  ?  What  effect  does  dividing  both  terms  of  a  fraction 
have  upon  it  ?  Why  ? 

Reduction  of  Fractions.  —  How  do  you  reduce  fractions  to  lower 
terms  ?  What  is  cancellation  ?  How  do  you  reduce  whole  or  mixed 
numbers  to  improper  fractions  ?  How  do  you  reduce  improper  frac- 
tions to  whole  or  mixed  numbers  ? 

Multiplication  of  Fractions.  —  How  do  you  multiply  a  fraction 
by  a  whole  number  ?  a  mixed  number  by  a  whole  number  ?  Explain,  by 
an  example,  the  method  of  multiplying  a  whole  number  by  a  fraction. 
Multiply  a  fraction  by  a  fraction ;  explain  and  give  the  rule.  How  do 
you  multiply  a  mixed  number  by  a  mixed  number  or  a  fraction  ?  How 


MISCELLANEOUS  EXAMPLES.  9X 

do  you  reduce  compound  fractions  to  simple  ones  ?  Can  you  give  one 
general  rule  for  multiplying  fractions,  whole  or  mixed  numbers,  by  frac- 
tions ? 

Division  of  Fractions.  —  How  do  you  divide  a  fraction  by  a  whole 
number  ?  a  mixed  number  by  a  whole  number  ?  a  whole  number  by  a 
fraction  ?  Explain,  by  an  example,  the  method  of  dividing  a  fraction 
by  a  fraction,  and  give  the  rule.  Give  one  general  rule  for  dividing  a 
fraction,  a  whole  or  mixed  number  by  a  fraction.  How  do  you  reduce 
complex  fractions  to  simple  ones  ?  How  do  you  find  what  part  of  one 
number  another  is  ? 

Least  Common  Multiple.  —  Define  a  multiple;  a  common  multiple 
of  two  or  more  numbers  ;  the  least  common  multiple.  When  do  you 
make  use  of  the  L.  C.  M.  ?  Give  and  explain  the  first  method  of  find- 
ing it;  the  second.  What  does  the  X.  C.  M.  of  prime  numbers 
equal  ? 

Common  Denominator.  —  When  are  fractions  said  to  have  a  com- 
mon denominator  ?  In  what  operations  upon  fractions  do  we  first 
reduce  them  to  those  having  the  same  denominator  ?  Can  we  change 
fractions  to  those  of  any  denominator?  How  ?  {Ans.  By  dividing  or 
multiplying'the  numerator  by  the  same  number  by  which  we  divide  or 
multiply  the  denominator  to  produce  the  denominator  required.) 
What  denominator  is  generally  chosen  ?  Reduce  a  simple,  a  compound, 
and  a  complex  fraction  to  those  of  the  same  denominator,  explain  the 
process,  and  give  the  rule. 

Addition  and  Subtraction  of  Fractions.- — How  do  you  add 
fractions  of  different  denominators  ?  How  do  you  subtract  one  frac- 
tion from  another  ?  How  do  you  add  mixed  numbers  ?  In  subtraction 
of  one  mixed  number  from  another,  how  do  you  proceed  when  the  frac- 
tion in  the  subtrahend  exceeds  that  in  the  minuend  ? 

G.  C.  D.  and  L.  C.  M.  of  Fractions^ —  How  do  you  find  the  G.  C. 
D.  of  fractions  ?  How  do  you  find  the  L.  C.  M.  of  fractions  ?  Find 
the  G.  C.  D.  of  I,  f  and  {^,  and  explain.  Find  the  L.  C.  M.  of  ^,  |, 
and  -^^,  and  explain. 

14:8,     Miscellaneous  Examples. 

1.  Into  strips  of  what  widths  may  I  cut  cloth  which  is  36 
inches  wide,  that  none  may  be  wasted,  the  width  of  the  strips  to 
be  expressed  in  inches  ? 

2.  How  many  gallons  in  the  largest  vessel  which  will  exactly 
measure  3  hogsheads,  containing  severally  128,  94,  and  158  gal- 
lons? 

*  Optinal. 


92  COMMON  FRACTIONS. 

a    What  will  16^  yards  of  cloth  cost  at  $.53  a  yard  ? 
4    What  cost  9^  bushels  of  corn,  at  $.87^  a  bushel  ? 
5.    What  cost  27 If  acres  of  land,  at  $31|  per  acre  ? 
6f  f  of  I  of  56  times  what  number  equals  ^§f  ? 

7.  I  paid  $.65  for  2  boxes  of  strawberries ;  what  will  be  the 
cost  of  45^  boxes  at  the  same  rate  ? 

8.  What  is  my  bill  for  7  pear  trees,  $1  apiece  for  the  trees, 
and  $2  a  dozen  for  setting  ? 

9.  What  do  I  receive  per  pound  by  selling  15f  pounds  of  cof- 
fee for  $3^? 

10.  -^  of  a  man's  property  is  in  land,  and  is  valued  at  $2324| ; 
what  is  the  value  of  his  whole  property  ? 

11.  Bought  f  of  an  acre  of  land  for  $40.75  ;  what  would  1 
acre  cost  at  the  same  rate  ? 

12.  What  costs  3  pieces  of  calico,  37^  yards  in  a  piece,  at  19^ 
cents  per  yard  ? 

13.  If  32y*2-  acres  of  land  cost  $1100,  what  costs  1  acre? 

14.  Sold  my  house  and  farm  of  47f  acres  for  $6150 ;  allow- 
ing $3500  for  the  house,  what  did  I  receive  per  acre  for  the* 
land? 

15.  How  long  will  a  barrel  of  flour  last  a  family  of  8  persons, 
if  it  lasts  3  persons  4^  months  ? 

16*  What  number  is  that  from  which  if  you  take  -^jj,  the  re- 
mainder will  be  ^V  ? 

17.  What  number  is  that  to  which  if  you  add  9|,  the  sum  will 
bel24|? 

18.  What  is  that  number  to  which  if  you  add  f  of  26^,  the 
sum  will  be  147-^? 

19.  Bought  7^  yards  broadcloth  at  $5  per  yard,  14^  yards  of 
kerseymere  at  $1;^  per  yard,  4|  yards  of  silk  at  $|  per  yard, 
and  f  yards  of  doeskin  at  $4^  per  yard,  for  w^iich  I  gave  in  pay- 
ment a  $100  bill.     What  balance  is  due  me  ? 

20r  I  have  paving  stones  12  inches  long  and  10  inches  wide; 
what  must  be  the  width  of  a  walk  which  will  just  receive  these 
stones,  laid  either  lengthwise  or  widthwise  ? 

21*  What  is  the  smallest  sum  of  money  which  can  be  exactly 
paid  in  pieces  of  money  worth  either  $.16f  or  $.12^? 


MISCELLANEOUS  EXAMPLES.  93 

22.  How  long  will  200  pounds  of  meat  last  9  persons  at  the 
t-ate  of  2|  pounds  a  day  for  each  person  ? 

23.  What  length  of  time  would  a  man  require  to  travel  around 
the  earth,  the  distance  being  25000  miles,  if  he  travel  at  the 
rate  of  31^  miles  per  day? 

24.  If  a  man  can  build  2f  rods  of  wall  in  a  day,  how  much 
can  he  build  in  6^  days  ? 

25.  What  is  that  number  J  of  which  exceeds  ^  by  2  ? 

Note.    ^  —  |  z=  ^  ;  if  ^  be  2,  |^  will  equal  20  X  2  =  40,  Ans, 

26.  What  number  is  that  f  of  which  exceeds  ^  by  llf  ? 

27.  How  many  bushels  of  wheat  can  a  man  purchase  for 
$2724^,  at  31f  cents  per  bushel? 

28.  What  is  f  -^  (yV  of  |  of  S^). 

29.  What  is  (§  of  ^)  -^  (^  of  f )  ? 

30.  If  I  buy  125  bushels  of  corn  at  41  §  cents  per  bushel,  and 
sell  it  at  521  cents  per  bushel,  what  do  I  gain? 

31.  AVhat  number  divided  by  |  equals  125^? 

32.  What  are  the  contents  of  3  floors  measuring  as  fol- 
lows :  13^  square  yards,  32Y^g  square  yards,  and  49||  square 
yards  ? 

33.  The  product  of  three  numbers  is  74^;  two  of  them  are  8f 
and  6^^^  ;  what  is  the  third  ? 

34.  Exchanged  42  tubs  of  butter,  averaging  48f  pounds,  at 
21 J^  cents  per  pound,  for  42  barrels  of  flour,  at  $9f  per  barrel,  and 
received  the  balance  in  cash ;  required  the  balance. 

35.  I  have  three  boxes  of  cloth,  each  containing  12  pieces, 
each  piece  containing  4f  yards,  weighing  3^  pounds  to  the  yard ; 
what  is  the  weight  of  the  whole  ? 

.    36.  What  will  42|  quires  of  paper  weigh  at  f  pound  per  quire? 

37.  A-grocer  has  five  casks  of  raisins  of  the  following  weights : 
115|  pounds,  117f  pounds,  9d^%  pounds,  100^  pounds,  and  121^ 
pounds  ;  what  is  the  average  weight  per  cask  ? 

38.  What  is  the  cost  of  the  above  at  8§  cents  per  pound  ? 

39.  Owning  f  of  a  paper-mill,  I  sold  f  of  my  share  for  $1750  ; 
what  is  the  value  of  the  whole  mill  at  the  same  rate  ? 


94  COMMON  FRACTIONS. 

40.  A  man  sold  50  yards  of  cloth  at  the  rate  of  Ij-  yards  for 
2  dollars  ;  what  did  he  receive  for  it  ? 

41t  Mr.  Gray  raised  212  bushels  of  potatoes,  f  of  which  he 
stored  in  4  equal-sized  bins  ;  what  did  each  bin  hold  ?  He  sold 
the  other  |  at  the  rate  of  3^  bushels  for  2  dollars  ;  what  did  he 
receive  for  the  potatoes  which  he  sold  ? 

42.  If  -If  of  a  ton  of  lead  cost  S134f,  how  many  ounces  of 
gold  at  SI  9^  per  oz.  will  pay  for  one  ton  of  lead  ? 

43.  When  hay  was  $15  per  ton,  I  gave  f  of  a  ton  for  If  tons 
of  coal ;  what  was  the  coal  worth  per  ton  ? 

44.  If  f  of  a  yard  of  cloth  will  pay  for  six  hats  worth  $4^ 
per  dozen,  what  is  the  price  of  the  cloth  per  yard  ? 

45.  If  /,y  of  an  acre  of  land  cost  $280^,  what  will  5f  acres 
cost? 

46.  If  a  man  walks  9^  miles  in  2^  hours,  how  far  will  he  walk 
in  4^  hours  ? 

47.  At  the  rate  of  4^  miles  an  hour,  what  time  Avill  be  re- 
quired to  walk  122  miles? 

48.  In  1860,  I  purchased  cotton  at  8^  cents  a  pound,  which  I 
sold  in  1862  at  90|  cents.     What  did  I  gain  on  1000  lbs.  ? 

49.  If  a  man  can  earn  $2^^^^  per  day,  how  many  days'  work 
will  he  have  to  give  for  a  suit  of  clothes,  of  which  the  coat  cost 
$25^,  the  pants  $8j\,  and  the  vest  $5^  ? 

50f  The  longest  canal  in  the  world,  is  the  Grand  Canal  in 
China;  ^  -f  ^  +  ^V  +  ^i^  +  tV  of  its  length  is  331§|| 
miles  ;  what  is  its  entire  length  ?     Ans.  650  miles. 

51.  If  f  of  I  of  a  ship  cost  $42,000,  what  is  f  of  her  worth? 

52.  In  a  certain  manufactory,  |  of  the  operatives  are  Germans, 
I  Dutch,  j\j-  Scotch,  ^  English,  -^^  Canadians,  and  the  remainder, 
140,  native  Americans  ;  what  is  the  whole  number,  and  the  num- 
ber of  each  nationality  ? 

53.  ^  of  my  money  is  in  gold,  ^  of  the  remainder  in  silver,  and 
the  balance,  $360,  is  in  bank  notes  ;  how  much  money  have  I  in 
all? 

54t  If  17  boxes  of  raspberries  cost  $2.83^,  what  part  of  a  box 
can  I  buy  for  12^  cents? 


^^■r' 


MISCELLANEOUS  EXAMPLES.  95 

55.  If  a  body  in  falling  descends  16^'^  feet  in  the  first  second 
()f  time,  three  times  16jL  in  the  next  second,  and  five  times  16^ 
teet  in  the  third  second,  how  far  will  it  fall  in  three  seconds  ? 

56.  Owing  a  man  in  Paris  1325^  francs,  I  have  shipped  to 
him  $375^  worth  of  rice.  If  the  franc  is  worth  18f  cents,  how 
much  liave  I  overpaid  him  in  Federal  money  ? 

57.  7f  oz.  of  gold  are  to  be  divided  among  3  men  and  a  boy, — 
the  boy  to  have  half  as  much  as  a  man  ;  what  will  each  have  ? 

58.*  If  the  wages  of  a  man  per  month  are  $35^,  and  if  the 
wages  of  3  boys  are  equal  to  the  wages  of  2  men,  what  will  be 
the  wages,  of  10  men  and  30  boys  for  a  month  ? 

59.  What  is  that  number  to  which  if  f  of  itself  be  added  the 
sum  will  equal  64  ?  Aris.  40. 

60r  If  from  5  times  a  certain  number  19  f  is  subtracted,  and 
the  remainder  is  18y^^,  what  is  the  number.'* 

61.  I  sold  my  watch  for  $72,  which  was  f  more  than  I  gave 
for  it;  what  did  it  cost  me  ? 

62.  Bought  a  horse  and  saddle  for  $75,  giving  |  as  much  fot 
the  saddle  as  for  the  horse ;  what  was  the  cost  of  each? 

63.  A  boy,  being  asked  the  age  of  his  dog,  replied,  "  If  ^  of 
his  age  be  added  to  his  age,  the  sum  will  be  13^  years."  What 
was  his  age  ? 

64.  Being  asked  the  age  of  his  father,  he  said,  "If  12  years 
were  added  to  ^  of  his  age,  the  sum  would  equal  ^  of  his  age." 
What  was  his  age  ?  Ans.  48  years. 

65.  Being  asked  his  own  age,  he  answered,  *'  If  2  years  were 
added  to  |  of  my  age,  the  sum  would  equal  ^  of  my  age.'* 
What  was  his  age  ?  Ans.  1 6  years. 

66.  A  can  build  a  wall  in  3  days,  and  B  can  do  the  same 
work  in  4  days.  What  part  of  the  work  can  each  do  in  one  day  ? 
What  part  can  both  do  in  one  day?  In  how  many  days  can 
both  do  it  working  together  ?  Ans.  If  days. 

67.  C  can  do  a  piece  of  work  in  5  days,  and  D  in  8  days. 
What  time  will  be  required  for  both  to  do  it  ? 

Ans.  3jJj  days. 


96  COMMON   FRACTIONS. 

68f  If  E  can  do  the  same  work  in  7  days,  how  long  would  be 
required  for  C,  D,  and  E,  to  do  it  wprking  together  ? 

Ans.  2yL8_  days. 

69t  If  A,  B,  and  C  can  do  a  piece  of  work  in  6  days,  and  A 
and  B  can  do  the  same  work  in  8  days,  in  what  time  can  C  do 
it  alone  ?  Ans.  24  days. 

70.*  Shipped  to  Havre  2000  bbls.  of  flour,  which  I  sold  at  $7| 
per  bhl. ;  received  in  return  oOOf  hhds.  of  wine,  worth  $21i  per 
■  hhd. ;  what  sum  is  still  due  me  ? 

7  It  A  merchant  owned  f  of  a  cargo  of  teas,  the  whole  cargo 
worth  $65000  ;  he  sells  f  of  his  share  for  $8583.331  y  does  he 
gain  or  lose,  and  how  much  ? 

72*  From  a  tank  containing  184  gallons  of  water,  20|^  gallons 
were  drawn  out ;  if  |  of  what  then  remained  was  equal  to  |  cf 
what  afterwards  rained  in,  how  much  rained  in?  How  much 
did  the  tank  then  contain?  Ans,  236|^-  gallons. 

73.*  I  pay  $700  for  a  piece  of  land ;  cut  52^  cords  of  wood 
from  it,  which  I  sell  at  $5.40  a  cord  ;  I  pay  $1^  a  cord  for  cut- 
ting and  hauling  the  wood,  and  $10  for  surveying  the  land ;  I 
divide  3  acres  of  it  into  house  lots  of  |  acre  each ;  4  of  these  I 
sell  at  $175  each,  and  the  rest  at  $162.50  per  lot.  Eeserving 
2  acres  for  myself,  valued  at  $300,  I  sell  the  remainder  of  the 
land  for  $600,  what  do  I  gain  ?  Ans.  $2388.18f. 

74t  Messrs.  B,  D,  W,  and  S,  built  a  drain  together,  each 
agreeing  to  pay  his  proportion  of  whatever  he  occupied.  B  oc- 
cupied 20  feet  alone,  B  and  D  22  feet,  B,  D,  and  W,  140  feet. 
B,  D,  W,  and  S,  18  feet.  The  drain  was  built  at  a  cost  of  33^ 
cents  per  foot ;  what  was  each  person's  share  of  the  cost  ? 

Note.  —  B's  share  ==  20  X  33 J -f-  22x_3_M  4-  lAJi^AM  +  LSx^^M, 

Ans.  B,  $27.38|  ;  D,  $20.72| ;  W,  $17.05|- ;  S,  $1.50. 
75t  A,  B,  C,  and  D,  hired  a  team  together  in  Boston  for  a 
journey  north,  each  agreeing  to  share  the  expense  for  the  dis- 
tance he  rode.  At  Reading,  14  miles  from  Boston,  A  got  out ; 
at  Andover,  8  miles  further,  B  got  out ;  at  Lawrence,  4  miles 
further,  C  left,  and  D  went  on  alone  8  miles  to  Haverhill.     Re- 


GENERAL   REVIEW.  9V 

turning,  he  took  up  C,  B,  and  A,  where  he  left  them,  and  all  rode 
into  Boston.  They  paid  $8.50  for  the  use  of  the  team ;  what 
was  each  one's  share  ? 

Note. — The  distance  from  Boston  to  Haverhill  is  34  miles ;  the  price  for 
I  mile  out  and  back  is  $ ^-^£-  =  $.25 ;  D's  share  is  JLA x 2_5 _^  _8_x^  ^  1.x 2^ 
+  8X25. 

Am,  A,  6.87^ ;  B,  $1.54^;  C,  $2.04^  ;  D,  $4.04^. 

14:0«    General  Review,  No.  3. 

1*  What  are  the  prime  factors  of  420  ? 

2,  Divide  15  X  7  X  12  X  «,  by  21  X  10  X  3  X  4. 

S,  What  is  the  greatest  common  divisor  of  21,  84,  and  51  ? 

4.  What  is  the  least  common  multiple  of  42,  9,  14,  and  12  ? 

5.  Reduce  ^f  f  and  -^-f^  to  their  lowest  terms, 

6.  Reduce  25 4f  to  an  improper  fraction, 

7.  Reduce  ^|^  to  a  mixed  number. 

8.  Reduce  y\  of  -^^  of  f  ^  of  6|  to  a  simple  fraction. 

9.  Reduce  |,  |,  and  f ,  to  a  common  denominator. 

10.  Reduce  ^f ,  ^f ,  and  8f ,  to  the  least  common  denominator, 

11.  Addfof^,  ij,  and9^. 

12.  Add  15|,  3^,  and  25^. 

13.  From  ^f  of  f^-  take  y\, 

14.  Subtract  S^^  from  10/^. 

15.  Multiply  y4  by  3|. 

16.  Divideloftkby  lOf 

17.  Change  _I_j  -f-,  -f,  and—  to  simple  fractions. 

18.*  Whaipartof  8f  is2i? 

19!  What  is  the  greatest  common  divisor  of  ^S  f,  and  2f  ? 
20*  What  is  the  least  common  multiple  of  ^*y?  fV?  ^^^  /f  • 
21*  How  many  fourths  of  |  of  40  in  3f  X  I  -^-  f  off*? 
I^  For  changes,  see  Key, 


i8  COMPOUND  DENOMINATE  NUMBERS. 

COMPOUND   DENOMINATE   NUMBERS. 

150 •  Numbers  are  either  Simple  or  Compound. 

151*  A  Simple  Ntunber  is  a  number  expressed  in  unitj  of 
one  denomination ;  as,  5  hoolcs^  7  pens. 

1«S3.  A  Compound  Number  is  a  number  expressed  in 
units  of  two  or  more  denominations,  but  of  the  same  nature ;  as, 
5  pounds  6  ounces  of  sugar ^  3  years  2  months  4  days  of  time, 

103.  Keduction  is  the  process  of  changing  the  denomi- 
nation of  numbers  without  altering  their  value. 

154*  Reduction  Descending  is  the  process  of  changing 
numbers  to  numbers  of  equal  value  in  lower  dentMninations ;  thus, 
1  dollar  =100  cents. 

\55^  Keduction  Ascending  is  the  process  of  changing 
numbers  to  numbers  of  equal  value  in  higher  denominations; 
thus,  100  cents  z=z  1  dollar. 

1«>6*  Compound  numbers  express  Currency,  Weight,  and 
Measure. 

Currency. 

Every  nation  has  its  own  currency.  That  of  the  United 
States  has  already  been  given  (Art.  68),  bnt  the  table  will  be 
inserted  here  for  the  sake  of  uniformity. 

157,    Federal  Money. 
The  denominations  are  eagles,  dollars,  dimes,  cents,  and  mills. 
The  legal  coins  in  circulation  are  as  follows : 


Gold. 

Silver. 

Double  Eagle 

=  $20.00. 

Dollar                   — 

$1.00. 

Eagle 

=:    10.00. 

Half  Dollar         = 

.50. 

Half  Eagle 

=:      5.00. 

'  Quarter  Dollar     =z 

.25. 

Quarter  Eagle 

=:      2.50. 

Dime                    = 

.10. 

Three  Dollar  piece 

=:      3.00. 

Half  Dime           = 

.05. 

One  Dollar  piece 

=      1.00. 

Three  Cent  piece  z=: 

,03. 

Copper  and  nickel  Cent,  and  Two  Cent  piece*. 


COMPOUND  DENOMINATE  NUMBERS.  99 

Note.  —  The  gold  coin  is  hardened  by  an  alloy  of  ^  copper  and  silver 
(the  silver  not  to  exceed  the  copper).  The  silver  coin  is  hardened  by  ^ 
copper.  The  cent  coined  since  1856  has  88  parts  of  copper  to  12  of  nickel. 
The  two-cent  piece,  coined  1864,  has  95  parts  copper  to  5  of  tin  and  zinc. 

Table. 

10  mills  (m.)  z=:  1  cent,  marked  c.  or  ct. 
10  c.                =r  1  dime,  ^*       d. 

10    d.  =11  dollar,        "       $. 

$10.  =1  eagle,         «       E. 

Note.  — Mill  is  derived  from  the  Latin  miller  one  thousand,  because 
1000  mills  =  1  dollar,  the  unit  of  computation,-  cent  from  Latin  centum, 
one  hundred,  because  100  cents  =  1  dollar;  dime  from  the  French  dime, 
a  tenth,  as  a  dime  is  one  tenth  of  a  dollar ;  dollar  from  the  German  thaler, 
dollar,  dollars  having  been  first  coined  in  Germany. 

Exercises. 

1.  Write  3  E.  $2.  7  d.  5  c.  2  m.  as  it  is  usually  written. 

Ans.  $32,752. 

2.  Write  162  E.  $8.  3  d.  9  c.  8  m.  as  it  is  usually  written. 

♦  Ans,  $1628.398. 


Write  in  the  same  manner, 

3.  128  E.  3  d.  8  m. 

4.  19  E.  $6.  3  c.  2  m. 

5.  68  E.  $8.  2  m. 


6.  $7.  2  c.  5  m. 

7.  $5.  6  d.  8  c.  3  m. 

8.  3984  E.  7  d.  4  c.  8  m. 


9.  Add  the  answers  of  the  last  six  examples,  and  give  the 
amount  in  mills.  Ans,  42,017,798  mill^ 

158,   English  Money. 

The  denominations  are  pounds,  sMlUngs^  pence^  Q,nd  farthings. 

Table. 

4  farthings  (qr.  or  far.)  =:  1  penny,    marked  d. 

12  d,  =1  shilling,        "       s. 

20  s.  =1  pound,  "      £. 

Note.  —  The  guinea  of  21  s.,  and  the  crown  of  5  s.,  are  also  used.    Th^ 
coin  which  represents  the  £  value  is  called  a  sovereign. 


12 

860  d. 

4 


lt)(>  REDUCTION  ASCENDING. 

109,    Eeduction  Descendino. 
Xll.  Ex.     Reduce  3  ^  11  s.  8  d.  2  far.  to  farthings. 

Operation.  ^^  20  s.  m  1  £,  we  shall  have  20  times 

3  £  11  s.  8  d.  2  far.       ^^  many  s.  as  £.       (20  X  3  )  s.  =:  60  s. ; 

20  60  s.  +  11  s.  =  71  s.     As  12  d.  zz:  1  s.,  we 

j~  shall  have  12  times  as  many  d.  as  s. ;  (12  X 

71)  d.  =  852  d. ;  852  d.  +  8  d.  =  860  d. 

As  4  far.  =i  1  d.,  we  shall  have  4  times  as 

many  far.  as  d. ;  (4  X  860)  far.  =  3440  far; 

3440  far.  +  2  far.  =  3442  farthings.  Hence 

3442  farthings,  Ans.       the 

EuLE  FOR  Reduction  Descending.  Multiply  the  number  of 
the  highest  denomination  by  the  number  which  it  takes  of  the  next 
lower  denomination  to  make  one  of  that  higher,  and  to  the  product 
add  the  given  number  of  the  next  lower  denomination.  Multiply 
that  sum  in  like  manner,  and  thus  proceed  till  the  number  is 
reduced  to  the  required  denomination. 

'  Examples. 

1.  Reduce  7  £  8  s.  3  d.  3  far.  to  farthings.  Ans.  7119  far. 

'     2.  Reduce  30  £  2  s.  0  d.  2  far.  to  farthings.      Ans.  28898  far. 

3.  Reduce  8  £  0  s.  3  d.  to  farthings. 

4.  Reduce  9  s.  Id.  2  far.  to  farthings. 

5.  Reduce  368  £  17  s.  2  d.  to  pence. 

6.  Reduce  25  crowns  3  s.  2  d.  to  farthiiags. 

7.  Reduce  43  crowns  4  s.  8  d.  to  pence. 

8.  Reduce  209  guineas  to  pence. 

9.  What  will  be  the  number  of  farthing  candles  that  may  be 
bought  for  2  s.  6  d.? 

160.    Reduction  Ascending. 

III.  Ex.  Reduce  3579  farthings  to  an  equivalent  value  in 
higher  denominations. 


^ 


REDUCTION.  101 

OPEftATioN.  As  4  qr.  =  1  d.,  we  shaU  have  1 

1  juuia  ^g  many  pence  as  farthings,  or  894  d. 

12  )  894  d.  +  3  qr.  and  3  qr.  remaining ;  as  12  d.  =r  1  s., 

20  )  74  s.  ■+-  6  d.  y^Q  gjiall  have  ^  as  many  shillings 

3  £  14  s.  6  d.  3  qr.,  Ans.  as  pence,  or  74  s.  and  6  d.  remain- 
ing. As  20  s.  1=  1  £,  we  shall  have  ^  as  many  £  as  s.,  or  3  £  and 
14  s.  remaining,  making  the  entire  result  3£  14  s.  6  d.  3  qr.  Hence 
the 

Rule  for  Eeduction  Ascending.  Divide  the  given  number 
by  the  number  which  it  takes  of  its  denomination  to  equal  one  of 
the  next  higher,  and  note  the  remainder.  Divide  the  quotient  thus 
obtained  as  before,  and  thus  -proceed  till  the  required  denomination 
is  attained.  The  last  quotient,  with  the  several  remainders,  will 
be  the  required  result. 

Proof.  As  reduction  ascending  is  the  converse  of  reduction 
descending,  either  process  may  be  proved  by  the  other. 

Examples. 

1.  Reduce  3681  farthings  to  an  equivalent  value  in  higher 
denominations.  Ans.  3  £  16  s.  8d.  1  qr. 

In  the  same  manner  reduce, 

2.  36875  farthings.  Ans.  38  £  8  s.  2  d,  3  far. 

3.  4328  pence.  Ans,  18  £  0  s.  8. 

4.  39818  shillings. 
o.  86347  farthings. 
6.  298721  farthings. 

161.  Comparison  of  English  and  Federal  Currency. 

1  £  =  $4.84. 


How  many  %  in 

1.  36  £?     ^?is.  $174.24 

2.  49  £? 

3.  64^  £? 


How  many  £  in 

4.  $39.43  ?  Ans.  8/3^  £. 

5.  $43.76? 

6.  $78.39  ? 


lot 


COMPOUND  DENOMmAVE  NUMBERS. 


WEIGHT. 

163.     Troy  Weight. 

Gold,  silver,  and  precious  stones  are  weighed  by  this  system. 
The    denominations   are   pounds,  ounces,  pennyweights,  and 
grains. 

Table. 

24  grains  (gr.)  z=z  1  pennyweight,  marked  pwt. 
20  pwt.  z=z  1  ounce,  "         oz. 

12  oz.  n:  1  pound,  "         lb. 


III.  Ex.      Reduce    2  lb.  9 
oz.  18  pwt.  3  gr.  to  grains. 
Operation. 

2  lb.  9  oz.  18  pwt.  3  gr. 
12 

33  oz. 
20 

678  pwt 
24 

2715 
1356 


III.  Ex.  Reduce  16275  gr. 
to  numbers  of  higher  denomina- 
tions. 

Operation. 
24  )  16275  gr. 


2|0)  67 1 8  pwt. -I- 3  gr. 

12  )  33  oz.  +  18  pwt. 

2  1b.+9oz. 
Ans.  2  lb.  9  oz.  18  pwt.  3  gr. 


16275  gr.,  Ans, 

Examples. 

1.  Reduce  18  lb.  11  oz.  5  pwt.  17  gr.  to  grains. 

Ans.  109097  gr.- 

2.  Reduce  48  lb.  2  oz.  0  pwt.  3  gr.  to  grains. 

3.  Reduce  1  oz.  23  gr.  to  grains. 

4.  Reduce  3681  lb.  9  oz.  1  pwt.  to  pennyweights. 
Reduce  to  equivalent  values  in  higher  denominations, 

5.  928641  pwts.  Ans.  3869  lb.  4  oz.  1  pwt. 

6.  3786541  grs. 

7.  9042028  grs. 

8.  What  is  the  value  of  2  lb.  8  oz.  of  gold,  at  $16  00  an 
ounce  ? 


REDUCTION. 


103 


9.  What  is  the  value  of  1  lb.  3  oz.  7  pwt.  of  gold,  at  4  cents 
per  grain  ? 

10.  What  will  20  silver  dollars  weigh,  each  dollar  weighing 
4121  grains.'^ 

163.     Apothecaries*  Weight. 

Apothecaries  use  this  weight  for  mixing  medicines ;  but  thejT 

buy,  and  generally  sell,  by  Avoirdupois  weight. 

The  denominations  are  pouyids,  omices,  drachms,  scruples,  and 

grains. 

Table. 

20  grains  (gr.)  =  1  scruple,  marked  se.  or  3. 

3  9  z=l  drachm,        "       dr.  or  5- 

8   5  ^=1  ounce,  "       oz.  or  §  - 

12  .§  z=  1  pound,  "        lb.  or  fc. 


III.  Ex.     Reduce  2  ife.  3  S, 
2  5,  1  9,  5  gr.  to  grains. 
Operation. 

21b 
12 

27  S 
8 

218  5 
3 

655  3 

20 


III.  Ex.  Reduce  68321 
grains  to  numbers  of  higher 
denominations. 

Operation. 
210)6832J1  gr. 


3)3416  9  4-lgr. 

8)1138  3  +  2  9. 

12)142  14-2  3. 

11  lb.  4- 10  §. 
Jbw.  lllb.4-10B,23,2  9,  Igr. 


13105  gr.,  Ans. 

Examples. 

1.  Reduce  5  lb.  7  i ,  75,  2  9,  12  gr.  to  grains. 

Ans.  32632  gr. 

2.  Reduce  3  lb.  0  S ,  7  5,  1  9,  9  gr.  to  grains. 

3.  Reduce  258481  grains  to  pounds,  ounces,  &c. 

Ans,  44  lb.  10  S,  4  3,0  9,  1  gr. 

4.  Reduce  36845  9  to  pounds,  ounces,  &c. 


104 


COMPOUND  DENOMINATE  NUMBERS. 


5.  Eeduce  987326  gr.  to  pounds^  ounces,  &c. 

6.  Reduce  28  ft.  3  S,  1  5,  2  9,  5  gr.  to  grains. 


Avoirdupois  Weight. 

ling  almost  all  articles,  except 


164. 

This  weight  is  used  for  wei| 
gold,  silver,  and  precious  stones. 

The  denominations  are  tons,  hundred  weighty  quarters y  pounds^ 
ounces,  and  drams. 

Table. 
16  drams  (dr.)=:l  ounce,  marked  oz. 

16  02.  :=1  pound,  "       lb. 

25  Ibw  =1  quarter,  "       q^. 

4  qr.  z=i\  hundred  weight,   "       ewt. 

20  ewt.  =1  ton,  «       T. 

Note.  —  The  long  ton  of  2240  lbs.,  which  gives  2S  lbs.  to  the  qr., 
is  sometimes  used  for  weighing  gross  articles,  as  iron  and  coal,  and  is  the 
ton  recognized  by  the  United  States  Government. 


III.  Ex.       Reduce  2  ewt.  3 
qr*  8  lbs.  to  pounds. 

Operation. 

2  ewt.  3  qrs.Slbs. 
4 

11  qr. 
25 

63 
22 

283  1b.,^ra*. 


III.  Ex.— Reduce  186421 
dr.  to  numbers  of  higher  de- 
nominations. 

Operation. 
16)186421  dr. 


16)11651  oz.4-5  dr. 

25)72?'lb.  +  3oz. 

4)  29  qr. 4-3  lb. 

7  ewt.  +  1  qr. 
Ans.  7  ewt.  1  qr.  3  \h.  3  oz.  5  dr. 


Examples. 

1.  Reduce  5  ewt.  3  qr.  24  lbs.  to  pounds. 

2.  Reduce  23  T.  4  lbs.  to  ounces. 


Ans.  599  lb& 


Reduce  to  equivalent  values  in  higher  denominations. 


8.  9328  lbs. 

Ans.  4  T.  13  ewt.  1  qr.  3  lb. 
i.  36842  oz. 


5.  193256  lbs. 

6.  8236548  dr. 

7.  9654321  dr. 


EEDUCTION.  105 

8.  Reduce  4  T.  3  cwt.  2  qr.  0  lb.  8  oz.  to  ounces. 

9.  How  many  ounces  in  1  cwt.  ?  in  1  T.  ? 

10.  How  many  pounds  in  one  ton  ? 

11.  How  many  more  pounds  in  a  long  ton  than  in  a  short  ton  ? 

12.  At  the  rate  of  3  lb.  a  day,  how  many  hundred  weight  of 
flour  will  a  family  consume  in  a  year,  or  365  days? 


165. 

Comparison  of  Weights. 

Troy. 

Apotii. 

Av. 

lib. 

=       lib. 

r^ 

Wlb. 

1  oz. 

==      1  § 

:z= 

Iffoz. 

Igr. 

=         Igr. 

=z 

T^oT^b. 

000  gr. 

=  7000  gr. 
Examples. 

^^^ 

1  lb. 

1.  Reduce  364  lbs.  Troy  to  Avoirdupois  weight. 

Ans.  299tV^  lbs.  Av. 

2.  Reduce  36  lbs.  Troy  to  Avoirdupois  weight. 

3.  Reduce  5  lbs.  Avoirdupois  to  grains  in  Apothecaries'  weight ; 
to  units  of  higher  denominations.  Ans.  6  lb.  0  § ,  7  3,  1  B. 

4.  Reduce  375  lbs.  Avoirdupois  to  Apothecaries*  weight. 

5.  Reduce  73  lbs.  Avoirdupois  to  Troy  weight. 

Measures  of  Extension. 
166.     Long  Measure. 
The  denominations  are  miles,  furlongs,  rods,  yards,  feet,  inches 
and  lines. 

Table. 

12  lines  (1.)  z=  1  inch,  marked  in. 
12  in.  ^Ifoot,        «       ft. 

3  ft.  =1  yard,       «      yd. 

5 J  yd.  or  16^  ft.  :=  1  rod,        "      r.  or  rd. 

40  r.  zzz.  1  furlong,  "       f.  or  fur. 

8  f.  =1  mile,       "      m. 


69^  miles  nearly         =  1  degree  (°)  of  longitude  at 

the  Equator, 
360  of  which  degrees  z=.  the  distance  round  the  earth. 

3  miles  zzz  1  land  league. 

1  mile  =  320  rods  =  5280  feet. 


106 


COMPOUND  DENOMINATE  NUMBERS. 


III.  Ex.     Reduce  2  m.  5.  f.  I      III.  Ex.     Reduce  1543514 
13  r.  4  yd.  2  ft.  to  inches.  I  inches  to  miles,  furlongs,  etc. 


Operation. 

m.  f.    r.    y.  ft. 
2.  5.*  13.  4.  2. 

8 

21  f. 
40 

853  r. 

426^ 
4269 

4695i 
3    yd. 


14088^  ft. 


169062  in,,  Ans. 


Operation. 
12  )  1543514  in. 

3  )  128626  ft.  +  2  in. 


5^  ==  J^  )  42875  y.  +  1  ft. 
2 

11  )  85750  halves  of  yd. 

410  )  77915  r.  +  I  yd.  -|  yds.  =  2  yd. 

1  ft.  6  in. 

8  )  194  f.  +  35  r. 

24  m.  +  2  f. 

24  m.  2.  f.  35  r.  2  yd.  1  ft.  2  in. 
1  ft.  6  in. 

Ans.  24  m.  2  f.  35  r.  2  yd.  2  ft.  8  in. 
Examples. 


1.  Reduce  3  m.  7  f.  14  r.  to  yards.  A7is.  6897  yds. 

2.  Reduce  3  f.  11  r.  2  yd.  1  ft.  7  in.  to  inches.  Ans.  26029  in. 

3.  Reduce  1590  inches  to  rods,  etc.  Ans.  8  rd.  0  yd.  0  ft.  6  in. 

4.  Reduce  5  m.  6  f.  3  r.  3  yd.  2  ft.  5  in.  to  lines. 

5.  Reduce  16906  inches  to  numbers  of  higher  denominations.^ 

6.  Reduce  1291968  lines  to  miles,  furlongs,  etc. 

7.  How  many  miles  round  the  earth  ? 

8.  How  many  miles  through  the  earth  from  pole  to  pole,  the 
distance  being  41704788  feet  ? 

9.  Find  the  cost  per  mile  for  grading  a  road,  at  90  cents  per 
rod.  Ans.  $288. 

10.  What  will  it  cost  to  fence  both  sides  of  a  road,  26  r.  2  yd. 
long,  at  $.65  per  yd.  ?  -  Ans.  $188.50. 

11.  How  many  furrows,  each,  10  in.  wide,  will  be  made  in 
ploughing  a  lot  of  land  lengthwise,  which  is  6  r.  1  ft.  wide  ? 


REDUCTION.  107 

167«     Surveyors'  Measure. 
The  denominations  are  miles,  chains,  rods,  links,  and  inches. 
Table. 
7y^5%  inches  z=z  1  link,  marked  1. 
25 1.  =1  rod,       «        r. 

4  r.  zz:  1  chain,    "        ch. 

80  ch.      =  1  mile,      «       m. 
1  chain  =  4  rods  =:  66  feet  =  100  links  =i  792  inches. 

Note.  —  Rods  are  seldom  used  by  surveyors,  the  distances  being  gen»- 
lally  taken  in  chains  and  links. 

III.  Ex.      Reduce  4  m.  75 1      III.   Ex.      Reduce    763218 
ch.  32  1.  to  links.  1  links  to  miles,  chains,  etc 

Operation.  Operatiok, 

4  m.  75  eh.  32  L  100 )  763218  ;  > 


80 


810)76312  ch-hl. 


395  ch. 

100  95:m. +  32ch. 


39532 1.,  Ans,  Am,  95  m.  32  ch.  18 1. 

Examples. 

1.  Reduce  3  m.  35  ch.  8  1.  to  links.  Ans,  27508  1. 

2.  Reduce  5  m.  78  ch.  2  1.  5  in.  to  inches. 

3.  13845  links  to  miles,  chains,  etc.  Ans.  1  m.  58  ch.  45 1. 

4.  259248  inches  to  miles,  chains,  etc. 

5.  In  1  m.  46  ch.  2  r.  how  many  rods  ? 

6.  In  9584  feet,  how  many  chains  ? 

168*     Mariners'  Measure. 
The    denominations    for    short    distances    are    cable-lengths, 
fathoms,  and  feet. 

Table. 

6  feet  (ft.)  z=z  1  fathom,  marked  fath. 
120  fath.       r=:  1  cable-length,  marked  c.  1. 
.  7|^  c.  1.        zzz\  common  mile,      "       m. 

Longer  distances  are  estimated  in  nautical  or  geographical 
miles,  each  mile  being  ^^  of  2^  degree  me^ured  on  a  great  circle 


108  COMPOUND  DENOMINATE  NUMBERS. 

of  the  earth,  and  averaging  6086.34  ft.*  or  1.15  -j-  common  milec 
3  nautical  miles  =  1  sea  league. 

Examples. 

1.  How  many  feet  in  7  c.  1.     32  fath.  ?  Am.  5232  ft 

2.  How  many  feet  in  5  c.  1.     4  ft.  ? 

3.  How  many  cable-lengths  in  672  fath.  ?  Ans.  5  c.  1.  72  fath. 
J.   Keduce  3684  feet  to  units  of  higher  denominations. 

169.     Cloth  Measure. 
Cloth  is  measured  by  its  length,  without  regard  to  its  width, 
^iie  yard  is  considered  the  unit  of  measure,  and  is  divided  into 
halves,  quarters,  eighths,  and  sixteenths. 

170.     Square  Measure. 

This  measure  is  used  for  determining  the  area  or  content*  of 
surfaces. 

The  denominations  are  square  miles,  acres,  roods,  square  rods, 
square  yards,  square  feet,  and  square  inches. 

Table. 

144  square  in.  (sq.  in.)  ■=.  1  square  foot,  marked  sq.  ft 
9  sq.  ft.  z=z\  square  yard,      "       sq.  yd. 

30J  sq.  yd.  or  272|^  sq.  rt.  =r  1  square  rod,        "       sq.  r. 
40  sq.  r.  =1  rood,  "       R. 

4  R.  z=il  acre,  "       A. 

640  A.  rz:  1  square  mile,      "       sq.  m. 

III.  Ex.     Reduce  2  sq.  m.  87  A.  3  R.  19  r.  to  rods.  . 

Operation. 

2  sq.  m.  87  A.  3  R.  19  r. 
640 

1367  A. 
4 

5471  R. 
40 


218859  r.,  Ans, 

Topographical  Bureau,  Washington,  2864. 


REDUCTION. 


109 


III.  Ex.     Reduce  386060  sq.  in.  to  square  rods,  yards,  &c. 
Operation. 
144  )  386060  sq.  in. 


9  )  2680  sq.  ft.  +  140  sq.  in. 

30J  )  297  sq.  yd.  +  7  sq.  ft. 
4  4 

121)   1188 

9  sq.  rd.  +  24|  sq.  yd. 
I  sq.  yd.  —  6  sq.  ft.  108  sq.  in.  (Art.  198.) 
9  sq.  rd.  24  sq.  yd.  6  sq.  ft  108  sq.  in. 
7  sq.  ft.  140  sq.  in. 

Ans.  9  sq.  rd.  25  sq.  yd.  5  sq.  ft.  104  sq.  in. 

r.  Reduce  3  sq.  m.  35  A.  to  square  yards.     Ans.  9462200  yd. 
2.  Reduce  19  A.  2  R.  5  sq.  rd.  to  square  inches. 
Reduce  to  numbers  of  higher  denominations, 


3.  9687  sq.  rd. 

Ans.  60  A.  2  R.  7  r. 

4.  5652  sq.  yd. 


5.  32865  sq.  ft. 

6.  84791  sq.  in. 

7.  932485  sq.  in. 


8.  What  are  2  A.  3  R.  15  r.  3  y.  8  ft.  of  land  worth  at  5  cents 
a  foot  ? 

171,     A  rectangle  is  a  figure  whose  opposite  sides  are  equal, 
and  whose  angles  are  right  angles.     (Art.  191.) 

17^.     A  square  is  a  rectangle  whose  sides  are  all  equal. 
173*    The  area  of  a  rectangle  is  found  by  rnultiplying  its 
length  hy  its  breadth. 

Illustration  I.  Suppose  the  length 
of  the  figure  A,  B,  C,  D,  to  be  4  inches, 
and  its  breadth  3  inches.  By  dividing 
the  line  B  C  into  4  equal  parts,  and  C  D 
into  3  equal  parts,  and  drawing  lines 
from  the  points  of  division  as  in  the 
figure,  it  will  readily  be  seen  that  the 
entire  figure-  is  divided  into  4  X  3,  or  12 


A  D 

I \ c 


1  inch. 


equal  parts,  each  part  containing  1  square  inch. 


110  COMPOLND  DENOMIl^ATE  NUMBERS. 

Illustration  II.  A  figure  1  in.  long  and  1  in.  wide  contains  1 
eq.  in.  A  figure  4  inches  long  and  1  inch  wide  must  contain  4  times 
as  many  sq.  inches,  or  4  sq.  inches.  A  figure  4  inches  long  and  3 
inches  wide  must  contain  3  times  as  many  sq.  inches  as  if  it  were  only 
1  inch  wide,  or  4  X  3  sq.  inches. 

The  area  of  a  rectangle  being  found  by  multiplying  its  length 
by  its  breadth,  it  follows  that 

When  the  area  and  one  dimension  of  any  rectangle  are  given, 
the  other  dimension  may  be  found  by  dividing  the  area  by  the  given 
dimension. 

Note. — ^In  performing  this  operation,  express  the  dividend  in  the 
superficial  denomination  corresponding  to  the  linear  denomination  of  the 
divisor ;  that  is,  if  the  divisor  is  expressed  in  feet,  the  dividend  must  be 
expressed  in  square  feet ;  if  in  yards,  the  dividend  must  be  expressed  in 
square  yards,  &c. 

174,     Examples. 

1.  If  one  side  of  a  rectangular  field  is  16  r.  7  ft.,  and  the  other 
12  r.  5  ft.,  how  many  square  feet  does  it  contain? 

2.  If  one  side  of  a  square  field  is  4  r.  8  ft.,  how  many  square 
feet  does  it  contain  ? 

3.  If  a  rectangular  field  measures  24  r.  2  fit.  in  length,  and 
17  r.  4  yd.  in  breadth,  how  many  square  yards  does  it  contain  ? 

4.  If  a  floor  contains  36  square  yards,  and  its  length  is  18  ft., 
what  is  its  width  ?  Ans.  18  ft, 

5.  If  a  ceiling  contains  30  6|^  sq.  ft.,  and  its  width  is  17^  ft., 
what  is  its  length  ? 

6.  A  garden  containing  f  of  an  acre  measures  on  one  side  192f 
feet  ;  required  the  length  of  the  other  side. 

7*  How  many  square  feet  and  inches  does  the  top  of  a  table 
contain,  which  measures  3  ft.  2^  in.  by  4  ft.  8  in.  ? 

8.  How  many  square  yards  of  carpeting  will  be  required  to 
cover  a  floor  17  feet  in  length  by  13  feet  in  width  ? 

9.  What  is  the  cost  of  oil-cloth  to  cover  a  floor  12  feet  by  16,^ 
feet,  at  75  cents  per  square  yard  ? 


DEDUCTION. 


Ill 


175J.     Solid  or  Cubic  Measure. 

This  measure  is  used  in  finding  the  contents  of  solid  bodies  or 
space,  i.  e.,  of  anything  that  has  length,  breadth,  and  thickness; 
height  or  depth. 

The  dimensions  are  cubic  yards,  cubic  feet,  and  cubic  inches. 

Table. 
1728  cubic  inches  (cu.  in.)  =  1  cubic  foot,  marked  cu.  ft. 
27  cubic  feet,  :=:  1  cubic  yard,       "       cu.  yd. 

Ncra.  —  The  denomination  ton  is  sometimes  used,  but  its  value  is 
variable,  a  greater  number  of  feet  being  assigned  to  the  ton  for  light 
bulky  articles  than  for  the  heavier. 

In  measuring  firewood  and  some  other  merchandise,  the  de- 
nomination cord  is  used.  A  pile  of  wood  4  feet  wide,  4  feet 
high,  and  8  feet  long,  contains  1  cord.  A  pile  4  feet  wide,  4  feet 
high,  and  1  foot  long,  contains  1  cord  foot.     Hence, 


16  cu.  ft. 

8  cd.  ft. 

or  128  cu.  ft. 


rz:  1  cord  foot,  marked  cd.  ft. 
=  1  cord,  "      cd. 


1  cord. 


led.  ft. 


J  cord, 

Examples. 

1.  In  3  cu.  yd.  18  cu.  ft.  136  cu.  in.  how  many  inches  ? 

Arts.  171,208  cu.  in. 

2.  Reduce  5  cu.  yds.  8  cu.  ft.  736  cu.  in.  to  inches. 

3.  Reduce  368742  cu.  in.  to  cubic  yards,  feet,  &c. 

Ans.  7  yd.  24  ft.  678  in. 

4.  Reduce  3427948  cu.  in.  to  cubic  yards,  feet,  &c. 

5.  How  many  cord  feet  in  36  c.  5  cd.  ft.  ? 

6.  How  many  cords  in  54328  cu.  ft.  ? 


112 


COMPOUND  DENOMINATE  NUMBERS. 


176*  A  solid  bounded  by  six  equal  squares  is  called  a 
Cube.  The  squares  are  called  the  faces  of  the 
cube,  and,  together,  make  its  surface.  The  bound- 
ing lines  are  called  edges.  If  its  edges  are  1  inch 
long,  it  contains  1  cu.  in. ;  if  1  foot  long,  it  contains 
1  cu.  ft.,  &c. 

177,  A  solid  that  is  bounded  by  rectangles  is  called  a 
Kectangular  Parallelepiped ;  rectangular,  because  its  faces  are 
rectangles,  and  parallelopiped,  because  its  opposite  faces  are  parallel. 

178,  The  Solidity  of  a  Parallelopiped  equals  the  product 
of  its  three  dimensions. 

Illustration.  Let  the  figure 
A  B  represent  a  parallelopiped  4 
.  feet  long,  2  feet  wide,  and  3  feet 
I  high.  If  it  is  4  feet  long  and  2  feet 
wide;  its  lower  face  or  base  must 
contain  4  X  2  zz:  8  square  feet.  If 
upon  these  square  feet  the  solid  ex- 
tends 1  foot  high,  it  will  contain  8 
cubic  feet  resting  upon  the  base. 
But  the  solid  is  3  feet  high,  and  must,  therefore,  contain  three  times  as 
many  cubic  feet  as  if  it  were  only  1  foot  high,  or  3  X  4  X  2  cubic  feet 
=  24  cubic  feet. 

Examples. 

1.  If  a  solid  is  3  ft.  long,  5  ft.  wide,  and  2  ft.  high,  how  many 
cubic  feet  does  it  contain  ?  Ans.  30  cu.  ft. 

2.  How  many  cubic  inches  in  a  block  3  in.  wide,  4  in.  high, 
and  1  ft.  2  in.  long?  Ans.  168  cu.  in. 

3.  How  many  cords  in  a  woodpile  40  ft.  long,  4  ft.  wide,  and 
4  ft.  high? 

179,  If  the  solidity  of  a  parallelopiped  equals  the  product 
of  its  three  dimensions,  it  follows  that 

When  the  solid  contents  and  two  dimensions  are  given,  the  third 
can  he  found  hy  dividing  the  contents  hy  the  product  of  the  two 
given  dimensions. 

When  the  solid  contents  of  a  block  and  the  area  of  its  base  are 
given,  how  do  you  find  its  height  ?  When  its  contents  and  height 
are  given,  how  can  you  find  the  area  of  its  base  ? 


4  feet. 


REDUCTION.  113 

4.  How  high  must  a  box  be  made,  to  contain  24  cu.  ft.,  the 
length  of  the  box  being  4  ft.  and  its  width  3  ft.  ?  Ans.  2  ft. 

5.  How  high  must  it  be,  if  its  length  is  8  ft.  and  its  breadth  3  ft.  ? 

6.  If  its  height  is  2f  feet,  what  must  be  the  area  of  its  base  ? 

7.  How  long  must  a  pile  of  wood  be,  which  is  4  ft.  wide,  3  ft. 
6  in.  high,  to  contain  a  cord  ? 

8.  There  are  144  square  inches  on  one  side  of  a  block  con- 
taining a  cubic  foot ;  what  is  the  length  of  the  edge  of  the  block  ? 

9.  There  being  112^  cubic  feet  in  a  stick  of  timber  which  is 
1 J  feet  square  at  the  end,  what  is  the  length  ?  Ans.  50  ft. 

Measures  of  Capacity. 
180*     Liquid    Measure. 
The  denominations  are  gallons,  quarts,  pints,  and  gills. 
Table. 
4  gills  (gi.)  =:  1  pint,      marked  pt. 
2  pts.  z=z  1  quart,  "       qt. 

4  qts.        ,  =  1  gallon,        "      gall. 
Note.  —  The  denominations  tierce,  barrel,  hogshead,  pipe,  butt,  and  ton, 
are  sometimes  used,  but  their  size  is  variable.    Barrels  generally  contain 
31-^  or  32  gall, ;  hogsheads,  63  gall. 

Casks  are  generally  gauged  and  marked  accordingly.    They  are  called 
hogsheads,  pipes,  butts,  or  tuns,  without  distinction. 

Examples. 

1.  Reduce  3  gall.  3  qt.  1  pt.  2  gi.  to  gills.  Ans.  126  gi. 

2.  Reduce  5  gall.  1  qt.  0  pt.  3  gi.  to  gills. 

3.  Reduce  23684  gills  to  gallons.  Ans.  740  gall.  1  pt. 

4.  Reduce  984324  gills  to  hogsheads. 

5.  What  will  27  gall.  3  qt.  of  milk  cost  at  4  cents  per  qt.  ? 

181.   Dry  Measure. 
The  denominations  are  bushels,  pecks,  quarts,  pints,  and  giUs, 
Table. 
4  gills  (gi.)  zzz.  1  pint,     marked  pt. 
2  pts.  zzz  1  quart,  '^      qt. 

8  qts.  zzz  1  peck,  "      pk. 

4  pks.         =  1  bushel,        "      bu. 
8 


114  COMPOUND  DENOMINATE  NUMBERS. 

Examples. 

1.  Redu«e  5  bu.  3  pk.  3  qt.  1  pt.  to  pints.  Ans.  375  pt. 

2.  Reduce  2641  pt.  to  bu.,  etc.         Ans.  41  bu.  1  pk.  0  qt.  1  pt. 

3.  Reduce  10  bu.  1  pk.  2  qt.  0  pt.  3  gi.  to  gills. 

4.  Reduce  8765432  gi.  to  bu.,  etc. 

5.  "What  will  4  bu.  1  pk.  2  qt.  of  cherries  cost  at  8  cts.  per  quart? 

6.  Sold  3  bu.  3  pk.  5  qt.  of  peaches  for  $7.50 ;  what  did 
receive  per  quart  ? 

183.    Comparison  of  Liquid  and  Dry  Measures. 

Liq,  Meas.  Dry  Meas.  Cu.  in. 

1  qt.  =z       57|. 

1  gall.  =:       231. 

1  qt.       r=       67f 

1  bu.      =:  2150f. 

Examples. 

1.  I  have  a  dish  that  contains  2  cu.  ft. ;  how  many  quarts  of 
blackberries  will  it  hold?  Ans.  51^. 

2.  How  many  quarts  of  water?  Ans.  59 ff  qt. 
-3.  How  many  gallons  of  water  will  a  cistern  hold  that  is  3  ft. 

long,  3  ft.  wide,  and  2^  ft.  high? 

4.  How  many  bushels  of  apples  can  be  put  into  a  bin  8  ft. 
long,  3  ft.  2  in.  wide,  and  2  ft.  high? 

Circular  or  Angular  Measure. 
183.    This  measure  is  used  principally  in  astronomy,  geogra- 
'phy,  navigation,  and  surveying. 

184,  A  Circle  is  a  plane  surface  bounded  by 
a  line,  every  part  of  which  is  equally  distant  from 

\  a  point  within  called  the  centre. 

185.  The  bounding  line  is  called  the  Circum- 
ference of  the  circle.  Any  part  of  the  circumfer- 
ence is  called  an  Arc. 

Fig-  2.  186.    A  straight  line  passing  from  the  centre  of 

the  circle  to  the  circumference,  is  called  a  Radius 
(plural,  radii). 

187.    A  straight  line  passing  from  one  point  in 
the  circumference,  through  the  centre,  to  an  opposite 
yoint,  is  called  a  Diameter. 


REDUCTION.  115 

188.  The  circumference  of  any  circle  is  supposed  to  be 
divided  into  360  equal  parts,  called  Degrees,  each  degree  into 
60  Minutes,  and  each  minute  into  60  Seconds. 

Table. 

60  seconds  {")  z=z  1  minute,  marked ', 

60'  =  1  degree,  "       «. 

360  °  z=  1  circumference,         "       circ. 

189.  A  Semi-circumference  is  half  a  circumference,  a 
Quadrant  one  fourth,  and  a  Sextant  one  sixth.  A  Sign,  used 
only  in  astronomy,  equals  30°. 

Fig.  3.  190  •   An  Angle  is  the  opening  between 

'^  two  lines  which  meet  each  other.  The  point 
of  meeting  is  called  the  Vertex  of  the  angle. 
The  angle  in  the  annexed  figure  may  be  read^ 
"  the  angle  a  h  c,"  or  simply  "  the  angle  h."  An 
angle  is  measured  by  that  part  of  the  circum- 
ference of  a  circle  included  between  its  sides,  the  centre  of  the 
circle  being  at  the  vertex  of  the  angle  ;  thus, 

rig.  4.  In  fig.  4,  the  angle  defis  measured 

by  the  arc  mn ;  that  is,  if  the  arc 
mn  contains  70°,  the  angle  defia  aw 
angle  of  70°. 

191.  An  angle  which  includes 
or  |-  of  a  circumference,  is  a 
Right  Angle,  the  sides  of  which  are  said  to  be  perpendicular  to 
each  other ;  in  fig.  4,  the  angle  g  eh  is  a.  right  angle.  An  angle 
greater  than  a  right  angle  is  an  Obtuse  angle.  An  angle  less 
than  a  right  angle  is  an  Acute  angle;  hed is  a,n  acute  angle 
and  ^  e  c?  is  an  obtuse  angle. 

Note.  —  As  arcs  are  measurements  of  angles,  the  table  for  angular 
measure  is  the  same  as  the  table  for  circular  meastire. 

199.   Examples. 

1.  Reduce  148°  54'  18  '  to  seconds.  Ans,  536058  '. 

2.  Reduce  354°  0'  16"  to  seconds. 


116  COMPOtTND  DENOMINATE  NUMBERS. 

3.  Eeduce  53684"  to  numbers  of  higher  denominations. 

Ans.  14°  54'  44". 

4.  Reduce  359°  59'  59''  to  seconds. 

5.  Reduce  1  quadrant  to  seconds. 

6.  How  many  seconds  in  1  sextant  ? 

7.  How  many  minutes  in  a  sign  ? 

8.  Reduce  35467"  to  numbers  of  higher  denominations. 

Time  Measure.- 

193*     The  length  of  an  Astronomical  or  Sidereal  Day  is 

the  time  the  earth  takes  to  turn  once  upon  its  axis;  the 
length  of  a  Solar  Day  is  the  time  the  earth  takes  to  turn  so  as 
to  bring  the  sun  to  the  same  meridian  again.  The  solar  day  is 
divided  into  24  hours,  each  hour  into  60  minutes,  and  each  min- 
ute into  60  seconds. 

The  denominations  of  time  are  centuries,  years,  months,  weeks, 
days,  hours,  minutes  and  seconds. 

Table. 


52  w. 


194*  The  time  which  the  earth  takes  to  revolve  around  the 
sun  is  365  d.  5  h.  48  m.  50  s.  nearly.  The  common  year  (365 
days)  thus  loses  nearly  one  day  in  4  years.  Hence  the  leap  year 
of  366  days  was  established,  which  occurs  once  in  4  years.  But 
this  adds  too  much  by  about  11^  m.  a  year,  which  in  100  years 
amounts  to  nearly  18§  h.  To  balance  this  error,  every  100th 
year  is  not  regarded  as  a  leap  year.  But  this  drops  too  much  by 
a  little  more  than  5]  h.,  which  in  4  centuries  amounts  to  nearly 
1  d.  Hence  every  four-hundredth  year  is  a  leap  year.  This  leaves 
an  error  which  is  less  than  1  d.  in  3600  years.     Hence  the 


60  seconds 

(8.) 

z=z  1  minute, 

marked  m. 

60  m. 

:=  1  hour. 

«        h. 

24  h. 

:=  1  day, 

d. 

7d. 

=  1  week, 

«                 -m 

d.  or  365  d. 

=  1  common  year, 

"         c.  y. 

366  d. 

r=  1  leap  year, 

1.  y. 

365^  d. 

:=:  1  Julian  year, 

"       J.  y. 

100  y. 

=i  1  century, 

C. 

REDUCTION. 


117 


Rule  for  ascertaining  when  any  year  is  a  leap  year. 
When  the  number  denoting  the  year  is  divisible  by  4,  and  not  by 
100,  it  is  a  leap  year  ;  and  any  year  that  is  divisible  by  400  is  a 
lea/p  year, 

]  0«5,  A  year  is  divided  into  four  seasons,  of  three  calendar 
months  each,  and  commences  with  January,  the  second  winter 
month. 

The  succession  of  th^  seasons ,  quarters  and  months,  and  the 
number  of  days  in  each  month,  are  shown  by  the  following 
diagram  : 

Quarter 


Quarter. 

«•  Thirty  days  hath  September, 
April,  June,  and  November  ; 
All  the  rest  have  thirty- one, 
Except  February  alone, 
To  which  we  twenty-eight  assign, 
Till  leap  year  gives  it  twenty-nine." 

Note.     In  the  following  examples,  common  and  leap  years  are  u»der« 
Btood  unless  the  Julian  is  specified. 

*  Leap  year.  29  d. 


118 


COMPOUND  DENOMINATE  NUMBERS. 


III.  Ex.     Reduce  2  y.  7  w. 
4  d.  4  h.  33  m.  to  minutes. 


Operation. 
7  w.  4  d.  =  53  d. 

2  y.     53  d.  4  h.  33  m. 
365 

783  d. 
24 

3132 
1566 


18796  h. 
60 

1127793  m. 


III.  Ex.  Reduce  5387294 
minutes  to  numbers  of  higher 
denominations. 

Opkration. 
610)53872914  m. 

24  )  89788  h.  +  14  m. 


365)3741  d.  + 4  h. 

10  y.  +  91  d. 
2 

89  d. 
As  2,  at  least,  of  the  10  years 
must  be  leap  years,  2  days  should 
be  taken  from  the  91  days  remain- 
ing, which  leaves  89  days. 

Ans.  10  y.  89d.  4h.  14  m. 
196.     Examples. 

1.  Reduce  8  y.  3  w.  19  d.  7  h.  to  hours,  allowing  for  2  leap 
years.  Ans.  71095  h. 

Note.      2  d.-f-  3  w.  19  d.=  42  d.,  .*.  the  example  may  be  stated.  Reduce 
8  c.  y.  42  d.  7  h.  to  hours. 

2.  Reduce  13  y.  8  w.  2  d.  3  h.  18  m.  to  minutes,  allowing  for 
3  leap  years. 

3.  Reduce  180739  hours  to  numbers  of  higher  denominations. 

Ans.  20  y.  225  d.  19  h. 
Reduce  5683762  minutes  to  numbers  of  higher  denomina- 


4. 

lions 

5. 

6. 


How  many  minutes  in  the  1st  century  ?   Ans.  52594560  m. 
How  many  hours  in  10  y.  36  d.,  beginning  with  Jan.  1st, 
1852?  Ans.  S853Qh. 

7.  How  many  seconds  in  the  3  summer  months  ? 

8.  How  many  days  from  April  12th,  1831,  to  May  3d,  1832  ? 
Note.  — From  April  12,  1831,  to  April  12,  1832  =  366  days;  to  May 

3,  21  days  more.     Ans.  387  days. 

9.  How  many  days  from  Jan.  1st,  1832,  to  Jan.  1st,  1863  ? 

10.  How  many  days  from  March  1st,  1850,  to  Jan.  1st,  1864? 

11.  How  many  seconds  in  10  years,  3^6  minutes,  allowing  365| 
days  to  the  year  ? 


A  book  formed  of 
sheets  folded 


KEDUCTION. 

197.     Miscellaneous  Table. 

Numbers.  \ 

12  units  or  single  things  :=  1  dozen. 
12  dozen  z=z  1  gross. 

12  gross  ==:  1  great  gross. 

20  units  or  single  things  ::==  1  score. 

Paper. 
24  sheets  of  paper  nz  1  quire, 
20  quires  z=  1  ream. 

in  2  leaves,  is  a  folio, 
in  4  leaves,  is  a  quarto, 
in  8  leaves,  is  an  octavo, 
in  12  leaves,  is  a  duodecimo  or  12mo, 
in  16  leaves,  is  a  16mo. 
in  18  leaves,  is  an  18mo, 
in  24  leaves,  is  a  24mo. 
in  32  leaves,  is  a  32mo. 
^  in  64  leaves,  is  a  64mo. 

Height  of  Animals. 

3  in.  =  palm. 

4  in.  :n  hand. 
9  in.  z=  span. 

Capacity. 
1  barrel  of  flour  =  196  lbs. 
1  barrel  of  pork  z=z  200  lbs. 

Examples. 

1.  How  many  rows  of  buttons,  6  in  a  row,  are  there  in  a  great 
gross  of  buttons  ? 

2.  In  3  score  and  6  years  how  many  days  ? 

3.  How  many  sheets  of  paper  in  3  reams,  7  quires,  21  sheets  ? 

4.  How  high  must  a  doorway  be  for  a  horse  to  pass  freely  un- 
der that  is  151  hands  high? 

5.  How  many  loaves  of  bread  can  be  made  from  a  barrel  of 
flour,  allowing  12|  oz.  to  the  loaf? 

6.  If  pork  is  worth  $18.75  a  bbl.,  what  is  it  worth  per  lb.? 


120  COMPOUND  DENOMINATE  NUMBERS. 

Suggestion.  The  pupil  may  now  write  from  memory  and 
present  for  inspection,  or  repeat  forward  and  backward,  the  table 
of  Federal  Money;  of  English  Money;  of  Troy  Weight;  of 
Apothecaries'  Weight ;  of  Avoirdupois  Weight ;  of  Dry  Measure ; 
of  Liquid  Measure ;  of  Long  Measure ;  of  Mariners'  Measure  ;  of 
Surveyors'  Measure ;  of  Square  Measure ;  of  Cubic  Measure ; 
of  Circular  Measure ;  of  Time. 

IS^  For  Dictation  Exercises  in  Reduction,  see  Key. 

FRACTIONAL  APPLICATIONS. 

198.   Reduction  of  a  Fraction   of   one   Denomination 
TO  Whole  Numbers  of  Lower  Denominations. 

III.  Ex.,  I.     Reduce  |£  to  shillings,  &c. 
Operation. 
f  £  =  I  of  20  s.  r=;  ^o  s.  =  13^  s. 
1  s.  r=  ^  of  12  d.  =  4  d.     Ans.  13  s.  4  d. 

III.  Ex.,  II.     Reduce  f  cwt.  to  quarters,  pounds,  &c. 

Opekation. 
6  cwt.  =  f  of  4  qr.  r=:  -\o  qr.  r=  2f  qr. 
f  qr.  z=  6  of  25  lb.  z=.  Ifii  lb.  =  21f  lb. 
•f  lb.  =  f  of  16  oz.  —  Y-  oz-  =  6f  oz. 
f  oz.  =  f  of  16  dr.  =  -\6.  (Jr.  =  13f  dr. 

Ans.  2  qr.  21  lb.  6  oz.  13|  dr. 

Or,  expressing  the  work  in  an  abbreviated  form, 

I  cwt.  z=  Y  qr.  =  2f  qr. 

4qr.  =:JL5o  lb.  — 21f  lb. 

f  lb.  =  -V  oz.  =  e^  oz. 

f  oz.  z=z  -^6.  dr.  =  13f  dr. 
Hence  the 

Rule.  To  reduce  a  fraction  of  one  denomination  to  whole 
numbers  of  lower  denominations :  Multiply  the  fraction  hy  the 
number  which  it  takes  of  the  next  lower  denomination  to  make 
one  of  that ;  reduce  the  fraction  thus  obtained  to  a  whole  or 
mixed  number,  if  possible.  If  a  fraction  remain,  proceed  with  it 
«5  before,  and  thus  continue  as  far  as  desired. 


FRACTIONAL   APPLICATIONS. 


121 


Examples. 
Reduce  to  whole  numbers  of  lower  denominations, 


1.  I  of  1  £.     Ans,  16  s,  8  d, 

2.  J  of  1  lb,  Troy, 

Ans,  10  oz.  10  pwt, 

3,  t  of  1  lb, 

4.  1^  of  1°. 

5,  1^  of  1  cwt. 
C.  I  of  1  c.  y, 

7.  -^2  ^^  1  gallon, 

6,  I  of  1  bu. 


9.  ^y  of  1  mile. 

10.  1^  of  1  furlong. 

11.  y^j  of  1  chain. 

12.  f  of  1  league. 

13.  ^2  ^^  1  s<l*  mile. 

14.  -^^  of  1  cu.  yard. 

15.  1^  of  1  cord. 

16.  f  of  IJ.  y, 

17.  ^  of  I  of  1  A. 


C^  For  Dictation  Exercises,  see  Key. 

199.    Reduction  of  WhoiLe  Numbers  of  Lower  Dexomi- 
nations  to  the  fraction  of  a  higher  denomination. 
III.  Ex.,  I.     Reduce  5  s,  3  d.  3  qr.  to  the  fraction  of  a  £. 


Operation. 


3qr.  =  }d. 
3Jd.  =  -V-d,=:-^5.of  ^s.= 


5 


6^8,  =:  f  I  S.=  If  of  2^^  = 


17 

0^ 


A 


£  =  J  J  £,  Am* 


16X^0 

III.  Ex.,  II.     Reduce  7  oz.  6  pwt.  16gr.  to  the  fraction  of  a  lb. 
Operation, 


16  gr. 


I  pwt. 


^0 


6|  pwt.  =  5^0-  pwt,  =  3->^g0  oz.  =  ^  oz. 


7i0Z,  =  -2j2  0Z,  = 


3x^;^ 


lb.  =  -i|lb.,  ^ws. 


Hence  the 

Rule.  To  reduce  whole  numbers  of  lower  denominations  to 
the  fraction  of  a  higher  denomination  :  Reduce  the  number  of  the 
lowest  denomination  to  a  fraction  of  the  next  higher.  Annex  it  to 
the  number  of  that  higher  denomination^  and  change  the  mixed 
.  number  thus  obtained  to  an  improper  fraction^  Reduce  as  before, 
and  thus  continue  as  far  as  desired. 


122  COMPOUND  DENOMINATE  NUMBERS. 

Examples. 
Reduce 

1.  6  s.  3  d.  to  the  fraction  of  a  £.  Ans.  ^%  £. 

2.  3  p.  6  qt.  1^  pt.  to  the  fraction  of  a  bu.  Ans.  ff  bu. 

3.  1  qt.  0  pt.  1  gi.  to  the  fraction  of  a  gall. 

4.  1  §,  2  3,  2  9 ,  to  the  fraction  of  a  lb. 

5.  5  cwt.  1  qr.  16  lb.  10§  oz.  to  the  fraction  of  a  T. 

6.  6  fur.  2  r.  2  y.  1  ft.  to  the  fraction  of  a  m. 

7.  4  y.  0  ft.  4J-  in.  to  the  fraction  of  a  r. 

8.  2  r.  1  1.  to  the  fraction  of  a  ch. 
What  part  of 

9.  1  A.  is  2  R.  1  r.  24  sq.  y.  6  sq.  ft.  108  sq.  in.  ? 

10.  1  cu.  yd.  is  13  cu.  ft.  864  cu.  in.? 

11.  1  cd,  is  5  cd.  ft.  4  cu.  ft.  576  cu.  in.? 

12.  1  c.  y.  is  162  d.  5h.  20  m.? 

13.  1  1.  y.  is  146  d.  9  h.  36  m.? 

14.  1  J.  y.  is  350  d.  15  h.  21  m.  3Q  s.? 

15.  If  1  £  is  worth  $4.84,  what  is  the  value  of  4  s.  6  d.  ? 
Solution.     4  s.  6  d.  =:  ^^^  £.     1  £  =2  $4.84, .  • .  ^  £  —  ^^  of 

$4.84 1=  $1,089. 
What  cost 

16.  3  pk.  2  qt.  of  meal  at  $X^0  a  bu.  ? 

17.  2  qt.  1  pt.  of  kerosene  oil  at  $.52  a  gall.  ? 

18.  62  lb.  8  oz.  soap  at  $7.50  per  cwt.  ? 

19.  2  g,  1  3,  4  gr.,  quinine  at  $4.00  per  g? 

20.  How  long  will  it  take  a  man  to  travel  9  miles  at  the  rate 
of  3  m.  6  f.  26  r.  3  yd.  2  ft.  an  hour? 

21.  At  $60.00  an  acre,  what  cost  2  A.  3  R.  13^  sq.  rd.? 

22.  At  $9.00  a  ton,  what  cost  1  T.  5  cwt.  2  qr.  14  lb.  of 
coal  ?     (Long  ton.) 

23.  At  $198  a  lb.,  what  cost  10  oz.  10  pwt.  10  gr.  of  gold? 

24.  Tlie  weight  of  a  cubic  foot  of  water  being  62|^  lbs.,  how 
many  pounds  of  water  will  a  tank  contain  which  measures  9  ft.  6 
in.  by  8  ft.  8  in.,  and  is  6  ft.  9  in.  deep  ? 

25.  A  cubic  foot  of  granite  weighs  163  lbs.  5  oz. ;  what  is  the 
weight  of  a  block  3  ft.  2f  in.  by  2  ft.  4  in.  and  1  ft.  3  in.  thick? 

I^"  For  Dictation  Exercises,  see  Key, 


ADDITION.  123 

300.     Addition. 
Addition  of  Compound  Numbers  is  the  process  of  finding  a 
number  equal  in  value  to  two  or  more  given  compound  numbers. 
The  process  is  similar  to  addition  of  simple  numbers. 

III.  Ex.  What  is  the  sum  of  3  £  11  s.  6  d.  3  qr.,  4  £  7  s.  8  d. 
2  qr.,  7  s.  6  d.  2  qr.,  and  9  £  18  s.  ? 

Operatiox.  Writing  the  numbers,  pounds  under  pounds, 

£.     s.       d.     qr.     shillings  under  shillings,  &c.,  we  commence 

3  11      6      3      by  adding  the  numbers  in  the  farthings'  col- 

4  7      8      2       umn,  and  find  the  amount  =  7  qr.  rz:  1  d.  +  3  qr. 
7      6      2       Writing  3  in  the  farthings'  place,  we  add  the 

9     18  Id.  with  the  column  of  pence,  and  have  for  the 

■ amount,  21  d.  =r  1  s.  +  9  d.     Writing  9  in  the 

Ans.  18  £  4  s.  9  d.  3  qr.  ^^^^^,  ^^^^^^  ^^  ^^^  ^^^  j  ^^  ^j^j^  ^^^  shiUings, 

and  have  44  s.  =:  2  £  -j-  4  s.  Writing  4  in  the  shillings'  place  we  add 
the  2  £  with  the  column  of  pounds,  and  have  for  an  answer,  18  £  4  s. 
9  d.  3  qr.     Hence  the 

Rule  for  Addition  of  Compound  Numbers.  Write  the 
numbers  of  like  denominations  in  the  same  column,  and  commence 
in  adding  with  the  numbers  of  the  lowest  denomination.  Divide 
the  amount  hy  the  number  it  takes  of  that  denomination  to  make  one 
of  the  next  higher,  write  the  remainder  under  the  column,  and  add 
the  quotient  with  the  numbers  of  the  next  higher  denomination. 
Add  the  next  column  in  the  same  manner,  and  thus  continue  till 
all  the  numbers  are  added. 

SOI.     Examples, 
Add  the  following  numbers : 

CURRENCY. 


1. 

2. 

a. 

$ 

£.  s. 

d. 

qr. 

£. 

s.  d.  qr. 

49.703 

5  8 

4 

2 

206 

18  4  3 

8.47 

7  15 

3 

3 

29 

14  9"^  2 

.882 

19 

0 

2 

118 

7  10 

4.369 

16  4 

6 

13  7  1 

Ans.  30     7     2     3 


124  <JOMPOUND  DENOMINATE  NUMBERS. 


WEIGHTS. 

^. 

; 

5. 

6. 

lb.  oz.  pwt.  gr.    lb. 
3  6   7   2     5 

5  0   3   8     5 

6  9  16  21     4 
5  3  15        5 

oz. 

3 
10 
11 

3 

pwt. 

2 
18 
17 

0 

23 
13 
15 

7 

T.  cwt.  qr.  lb.  oz. 

5  16  3  20  8 

4   0  2  17  3 

18  0   5  4 

15   6  3   4  14 

7. 

8. 

9. 

t.  cwt  qr. 

18   5  1 

7  3 

9  16  0 

lb.  oz.  dr. 

22  14  8 

4  15  7 

15  6  14 

ft. 

5 
3 

18 

§.  3 

9  7 
5  2 
4  7 

.  B. 

'     2 
!  1 
'  2 

gr.   lb.  g.  3. 

18   38  11  6 

5      6  7 

17    5  9  4 

B. 

0 
1 
2 

4 
13 
15 

MEASURES  OF 

LENGTH. 

10. 

11. 

m.  f.  r. 
3  4  32 
7  7  38 
5  3  19 

yd.  fl. 
4   2 
0   1 
4   2 

deg. 

3 

8 

20 

m. 

28 
59 
17 

f.  r.  yd.  ft. 
7  36  4   0 

4  18  3   2 

5  37  4   1 

in. 

7 
5 
9 

1. 

9 

7 

11 

17  0  10 

3^  2 
i=l  6  in. 

32 

36t 

2  13  1   1 
=  6  26  3   2 

11 

3 

17  0  10 

4   0  6  in. 

32 

37 

0  39  5   0 

11 

3 

12. 

13. 

m.  f. 
5  4 

39  7 

40  6 

r.  yd.  ft. 

36  2   2 
28  3   1 
17  2   0 

14. 

in. 
7 
9 

11 

deg. 

58 
8 
9 

m.  f.  r.  yd. 
59  7  30  2 
47  6  31  3 
64  0  14  4 

15. 

ft. 
2 
1 
2 

- 

m.  ch.  r.  1.  in 

. 

m 

.  ch.  r.  1.  ] 

in. 

3    2  3  20  5 
8   73  2  19  4 
5   13  0  22  8 

1 
3 
2 

39  3  24 
0  1  18 
3  0  21 

7 
3 
5 

^ns.      m.      ch.  r,      1.  Iyq%  in. 


ADDITION. 

125 

16. 

17. 

c.l. 

fath. 

ft. 

c.l.    fath.     ] 

ft. 

10 

12 

2 

9      96 

5 

4 

94 

4 

12    102 

3 

11 

37 

5 

8      86 

2 

MEASURE    OF 

SURFACE. 

18. 

19 

'. 

A. 

R.     sq.  r. 

sq.y. 

sq.  ft. 

m.    A.     K. 

sq.  r.     sq.  ft. 

3 

3       33 

13 

8 

2       28     3 

29         147 

15 

2       16 

12 

7 

3     520     2 

36         208 

22 

1       27 

4 

6 

CUBIC    ME. 

5     361     3 

22         168 

ASURE. 

20. 

21. 

cu.  yd. 

cu.ft 

;.     cu.  in. 

cd.      cd.  ft. 

cu.ft. 

320 

20 

1000 

18         6 

13 

29 

24 

968 

27        7 

14 

oOO 

0 

728 

36         5 

15 

MEASURES    OF 

CAPACITY. 

22. 

23. 

gall.     qt. 

pt. 

gi- 

bu.     pk. 

qt.     pt. 

18       3 

1 

3 

185      3 

7       1 

5       3 

0 

2 

39      2 

5       1 

6       0 

1 

3 

98      0 

6      0 

8       2 

1 

1 

102      3 

1       1 

CIRCULAR 

MEASURE. 

TIME 

, 

24. 

25. 

o            / 

// 

y.*    d.      h. 

m.     s. 

31         4 

18 

2     328    18 

26     31 

37       30 

27 

5       27      7 

24     45 

27       24 

54 

3       79      6 

58     39 

128       44 

58 

281    23 

41     23 

*  Art.  195.    Note. 


126  COMPOUND  DENOMINATE  NUMBERS. 


26. 

27. 

y- 

w. 

d. 

h. 

y- 

w. 

d. 

h. 

m. 

7 

2 

8 

5 

17 

18 

5 

13 

48 

2 

47 

3 

0 

3 

49 

8 

7 

0 

8 

39 

2 

2 

4 

39 

6 

7 

18 

13 

37 

6 

1 

7 

Ans.  13       37       5       7  Ans.  26        4      4        4       6 

Note.  —  As  it  takes  52  w.  4"  1  d.  to  equal  one  y.,  for  every  year  that 
is  added  to  the  column  of  years,  1  day  must  be  taken  from  the  amount 
of  days. 

28.  Eequired  the  contents  of  3  hogsheads  containing  respec- 
tively 58  gall.  3  qt.,  67  gall.  2  qt.,  anti  89  gall.  3  qt. 

29.  Mr.  Rice  bought  cranberries  as  follows :  7  bu.  3  pk.  2  qt., 
2  bu.  1  pk.  7  qt.,  5  bu.  2  pk.  6  qt. ;   required  the  amount. 

30.  How  far  will  Mr.  Brown  travel  in  four  days,  if  he  travels 
on  the  1st  day,  25  m.  2  f.  7  r.  3  yd.,  on  the  2d,  18  m.  7  f.  38  r.  2 
yd.,  on  the  3d,  23  m.  2  f.  4  r.  6  yd.,  and  on  the  4th,  31  m.  5  f.  12  r.  ? 

31.  How  much  land  have  I  in  4  pastures,  the  1st  containing 
7  A.  2  R.  3  r.  31  yd.,  the  2d,  15  A.  3  R.  26  r.,  the  3d,  22  A.  1  R. 
12  r.  18  yd.,  and  the  4th,  5  A.  0  R.  9  r.  2  yd.  ? 

J^'  For  Dictation  Exercises,  see  Key. 

303*    Subtraction. 

Subtraction  of  Compound  Numbers  is  the  process  of  find- 
ing a  number  equal  to  the  difference  between  two  given  com- 
pound numbers. 

III.  Ex.     Subtract  2  bu.  3  pk.  7  qt.  from  5  bu.  2  pk.  2  qt. 
Operation.  As  we  cannot  take  7  qt.  from  2  qt.,  we  must 

bu.     pk.     qt.      reduce  1  of  the  2  pk.  in  the  minuend  to  quarts, 
2       3^         which  —  8  qt. ;  8  qt.  +  2  qt.  =  10  qt.  ;    7   qt. 

from  10  qt.  z=z  3  qt.,  which  we  write  under  the 

Ans.  2bu.  2  pk.  3  qt.  column  of  quarts.  There  is  but  one  peck  left  in 
the  minuend.  As  we  cannot  take  3  pk.  from  1  pk.,  we  must  reduce  1 
of  the  5  bu.  to  pk.,  which  =  4  pk. ;  4  pk.  +  1  pk.  =  5  pk. ;  3  pk. 
from  5  pk.  z=z  2  pk.,  which  we  write  under  the  column  of  pecks.  2  bu. 
from  4  bu.  =  2  bu.     The  answer  is  2  bu.  2  pk.  3  qt.    Hence  the 


SUBTRACTION.  127 

Rule  for  Subtraction  of  Compound  Numbers.  Write 
the  subtrahend  beneath  the  mimcend,  so  that  numbers  of  the  same 
denomination  shall  he  in  the  same  column.  Commence  with  the 
lowest  denomination  ;  subtract  each  number  from  that  immediately 
above  it^  writing  the  difference  beneath.  If  any  upper  number  is 
less  than  the  lower,  increase  it  by  adding  to  it  one  of  the  next 
higher  denomination  rednced  to  that  denomination,  and  then  sub- 
tract,  bearing  in  mind,  in  the  next  operation,  that  the  upper  number 
is  less  by  the  one  reduced. 

Proof.     The  proof  is  the  same  as  in  simple  subtraction. 
303*    Examples. 


1. 

2. 

3, 

£. 

From       18 
Subtract    5 

s.    d. 

7  6 

8  9 

18     8 

qr. 
2 
3 

3 

.     lb. 
9 
2 

oz.  ] 
6 
9 

^^i.  gr. 

18     13 

5     18 

ft.      §.     5.  9.  gr. 

15       7     6     0      9 

5     11     4     2    11 

Ans.  12 

4. 

5. 

6. 

ni.    f.   ch. 

r.      1. 

o 

4 

11 

cd.     cd.  ft.    cu.ft. 

1     1     8 
3     9 

3     17 
3     23 

7. 

35 

19 

47 
54 

28 
48 

50         3         12 
25         7         15 

8. 

y.*    d. 

h. 

m. 

s. 

y.*  w. 

d.    h.     m.     s. 

2     248 
1     324 

7 
18     i 

0 

3G 

19 

27 

7     37 

5     50 

4     2     12     18 
3     0     18     42 

9. 

10. 

11. 

gall.qt.  ] 
0     2 
4     3 

pt.  gi. 

1     2 

0     3 

12, 

bu.  pk.  qt. 
7     2     0 
3     5     2 

pt.           T.  cwt  qr.  lb.  oz. 

0  3     0     0     0     0 

1  1     7 

13. 

T.   cwt.  qr. 

lb. 

oz.    dr. 

cii.yd. 

.  cu.ft.  cu.  in. 

20 

3     0 
17     3 

2 

8 

13 

9    : 

•Art. 

10 
195. 

1860 
58 

Note. 

0           17 
2       1720 

128  COMPOUND  DENOMINAIE  NUMBERS. 

14.  15. 

rd.  yd.       ft.  in.  m.    f.  r.    yd.  ft.   in. 

32         19  26  30     323 

2     2         2     2  7  31     4     1     6 

4^       2     7 

Ans.  5         11 

16.  17. 

gq.m.  A.  R.  sq.r.  sq.y.  sq.ft.  sq.in.     sq.m.  A.  R.  sq.r.  sq.y.  sq.fl. 

6     .  5    1     18       8       5         18  3  7     2       0       0       0 

2      68    1     27     19       4        116  18     28       8 


18. 

19. 

c.  1.    fath.  ft. 

cd.     cd.  ft.  cu.  ft.  cu.  in. 

36       18     5 

394         4         7         59 

9       97     31- 

15         7       la       410 

S04.   III.  Ex.     Required  the  time  from  June  5,  1862,  to 

Jan.  1,  1863, 

y.       mo.     d-  Jan.  being  the  1st  m.  and  June  the  6th,  we  sub- 

1863       1         1       tract  1862  y.  6  m.  5  d.  from  1863  y.  1  m.  1  d.,  allow- 

iggo       6         5  J  ' 

..     ^ ing  30  days  for  one  month  and  12  months  for  a 

Ans.  6  m.  26  d.  year,  and  have  for  the  answer  6  m.  26  d. 

Note.  —  This  method  is  not  so  exact  as  that  illustrated  in  Art.  196,  hut 
is  one  in  common  use. 

20.  Required  the  time  from  the  landing:  of  the  Pilgrims,  Dec 
22,  1620,  to  the  Declaration  of  Independence,  July  4,  1776. 

21.  From   the  Declaration  of  Independence  to  the  present 
time. 

22.  Nathaniel  Bowditch  was  bom  March  26,  1773,  and  died 
March  16,  1838;  what  was  his  age? 

23.  Horace  Mann  was  born  May  4, 1796^  and  died  Aug.  2, 
1859  ;  what  was  his  age  ? 

24.  Lafayette  was   born    Sept.    6,  1757,  and   died  May   19, 
1834  ;  what  was  his  age  ? 

25.  Required  the  time  between  the  births  of  Tycho  Brahe, 
Dec.  4,  1546,  and  La  Place,  March  23,  1749. 


ADDITION  AND  SUBTRACTION.  129 

26.  Required  the  time  between  the  births  of  Benjamin  Frank- 
lin, Jan.  17,  1706,  and  George  Washington,  Feb.  22,  1732. 

27.  How  old  was  Washington  when  he  died,  Dec.  14,  1799  ? 

28.  How  old  was  Lafayette  when  that  event  took  place  ? 
j^*  For  Dictation  Exercises,  see  Key, 

300*   Addition  and  Subtraction  combined. 

1.  3  qr.  9  lb.  8  oz.  +  7  cwt.  9  lb.  2  oz.  +  28  cwt.  2  qr.  =  ? 

2.  91b,  9  §,15,  19,  9gr.  — 2tb,  11  §,  4  5,  2  9,  18gr.  =  ? 

3.  17£.  9s.  6d.  3qr.  +  2£.  12s.  8  d.  2qr.  — 7£.  15  s.  8  d. 
3qr.  =  ? 

4.  28  cd.  2  cd.  ft.  14  cu.  ft.—  19  cd.  6  cd.  ft.  2  cu.  ft.—  4  cd. 
5  cd.  ft.  14  cu.  h.  +  18  cd.  3  cd.  ft.  9  cu.  ft.  ? 

5.  1848  cu.  yd.  9  cu.  ft.  1700  cu.  in.—  (118  cu.  yd.  7  cu.  ft. 
176  cu.  in.  +  50  cu.  yd.  960  cu.  in.)  =  ? 

6.  19  lb.  5  oz.  12  pwt.  19  gr.  +  9  lb.  3  oz.  18  pwt.  2  gr.  +7 
lb.  8  oz.  17  pwt.  22  gr.—  (13  lb.  3  oz.  12  pwt.  18  gr.  —  2  lb.  12 
pwt.  16  gr.)  z=  ?  Ans.  25  lb.  3  oz.  8  pwt.  17  gr. 

7.  7  T.  5  cwt.  2  qr.  20  lb.  +  2  T.  8  cwt.  3  qr.  7  lb.  —  (2  T. 
7  cwt.  18  lb.  —  1  T.  19  cwt.  24  lb.  10  oz.)  z=  ? 

8.  3°  7'  18''  +  9°12'40"  +  40«  12' -[- 90°  — 7°  2'8"  —  (19° 
3/  10"  -|_  12°  3'  13")=? 

9.  9y.  22  w.  3d.  lOh.  — 4y.  7  w.  5  d.  3  h.-f8  y.  45  w.  3d. 
12h.  —  2y.  25  w.  6d.  2  h.  =  ? 

10.  8  m.  7  f.  12  r.  4  y.  1  ft.  +  4  f.  3  r.  +  4  m.  5  f.  30  r.  2  y. 
—  3  m.  7  f.  38  r.  5  y.  1  ft.  4  in.  =? 

11.  3  sq.  ra.  204  A.  5  sq.  r.  9  sq.  yd.  +  98  A.  3  R.  2  sq.  rd. 
4  sq.  yd.—  (140  A.  2  R.  34  sq.  rd.  28  sq.  yd.  +  278  A.  1  R.  39 
sq.  yd.)  1=  ? 

12.  384  cu.  yd.  19  cu.  ft.  1700  cu.  in.  —  (207  cu.  yd.  2  cu.  ft. 
18  cu.  in.  — 116  cu.  yd.  18  cu.  ft.  394  cu.  in.)  +  504  cu.  yd.  24 
cu.  ft.  89  cu.  in.  :=  ? 

13.  Mr.  Day  having  49£  ready  money,  pays  to  one  man  a  debt 
of  5£.  7  s.  8  d. ;  to  another,  10£.  15  s.  6  d.  to  the  third,  18£.  12  s, 
9  d. ;  how  much  money  has  he  left  ? 

9 


130 


COMPOUND  DENOMINATE  NUMBERS. 


306.    Table  of  Latitudes  and  Longittdes.* 

Longitude  from 
Place.  " 

Albany, 

Boston, 

Canton, 

Calcutta, 

Cape  Horn, 

Cape  of  Good  Hope, 

Charleston, 

Chicago, 

Cincinnati, 

Constantinople, 

London, 

Mexico, 

Montreal, 

New  Orleans, 

New  York, 

Paris, 

Philadelphia, 

Portland, 

Quebec, 

San  Francisco, 

St.  Petersburg, 

Washington, 

14.  What  is  the  difference  of  latitude 
Charleston  ? 

15.  Between  Washington  and  London  ? 
IG.  Between  Philadelphia  and  Paris? 

17.  Between  Quebec  and  St.  Petersburg  ? 

18.  Betw^een  Portland  and  Cape  Horn  ?  f 

19.  Between  Cape  Horn  and  Cape  Good  Hope? 

20.  What  is  the  difference  of  longitude  between  Boston  and 
Montreal  ? 

21.  Between  New  Orleans  and  New  York? 

22.  Between  Mexico  and  Paris  ?t 

*  From  the  American  Almanac  and  New  American  Cyclopedia. 

t  The  difference  of  latitude  between  places  on  opposite  sides  of  the 
equator  is  found  by  adding  the  latitudes.  The  difference  of  longi- 
gitude  between  places  on  opposite  sides  of  the  first  meridian  is  found  by 
adding  the  longitudes.  If  their  sum  exceeds  180°,  the  difference  of  longi- 
tude equals  360°  minus  that  sura. 


State. 

Greenwich. 

Latitude. 

N.  Y, 

w. 

73  44  39 

N. 

42  39  50 

Mass., 

w. 

71     3  30 

N. 

42  21  27 

China, 

E. 

113  14 

N. 

23     7 

India, 

E. 

88  19     2 

N. 

22  35     5 

S.  America, 

W. 

67  16     8 

S. 

55  58  40 

Africa, 

E. 

18  29 

S. 

32  24     3 

S.  C, 

W. 

79  55  38 

N. 

32  46  33 

111., 

W. 

87  37  47 

N. 

42     0     0 

Ohio, 

W. 

S4  27 

N. 

39     5  54 

Turkey, 

E. 

28  59 

N. 

41     0  16 

England, 

W. 

5  48 

N. 

51  30  48 

Mexico, 

W. 

103  45  30 

N. 

19  25  45 

L.  C, 

W. 

73  35 

N. 

45  31 

La., 

W. 

90 

N. 

29  57  30 

N.Y., 

W. 

74    0     3 

N. 

40  42  43 

France, 

E. 

2  20  22^ 

N. 

48  50  12 

Pa., 

W. 

75     9  54 

N. 

39  58  24 

Me., 

W. 

70  14  34 

N. 

43  39  54 

L.  C, 

W. 

71  12  18 

N. 

46  49  12 

Cal., 

W. 

122  26  48 

N. 

37  47  53 

Russia, 

E. 

30  19 

N. 

59  56  30 

B.C., 

W. 

77     0  15 

N. 

38  53  20 

between  Boston  and 
Ans.  9°  34'  54". 


DENOMINATE  FRACTIONS,  131 

23.  Between  Constantinople  and  Chicago  ? 

24.  Between  Canton  and  San  Francisco  ? 
1^  For  Dictation  Exercises,  see  Key. 

d07.  Addition  and  Subtraction  of  Denominate  Frac- 
tions. 
III.  Ex.,  I.     f  cwt.  -[-  f  qr.  =:  what .? 

These  fractions,  being  of  different  denominations,  must  first  ba  re- 
duced to  the  same  denomination.  The  fraction  |  cwt.  may  be  changed 
to  quarters,  or  the  f  qr.  may  be  changed  to  parts  of  a  cwt.,  and  then  ad- 
dition can  be  performed.  In  the  first  case,  the  answer  will  be  in  qr. ; 
in  the  second,  in  cwt. 

Another  excellent  method  is  first  to  reduce  both  fractions  to  integers  of 
lower  denominations,  if  necessary  (Art.  198),  and  then  add.     Thus, 
qr.     lb.      oz. 
fcwt.  =  2     16     lOf 
I  qr.  z=         18     12 

3     10       6f,     Ans, 
III.  Ex.,  II.     f  £  —  |  5.  =  what  ? 

%£~  ISs,  4d, 
Operation.     ^ 


12    6|,    Am, 
Examples. 
Perform  the  following  examples,  and  give  the  answers,  as  far 
as  possible,  in  whole  numbers  of  lower  denominations. 

1.  I  ch.  +  T-\  rd.  =  ?  Ans,  2  rd.  23/^  1. 

2.  f  y.  +  f  d.  =  ?*  Ans.   81  d.  8  h. 
3-  3U  J-  +  s'^d.=z?                              Ans.  80  d.  7  h.  20  m. 

4.  ^  lb.  -f  I  §  =  ?  A71S.  1  oz.  5  dr.  2  sc. 

5.  /g-  cu.  yd.  +  i  cu.  ft.  =  ?  Ans.  11  cu.  ft. 

6.  §  cd.  —  l  cd.  ft.  r=  ?      Ans.  5  cord  ft.  1  cu.  ft.  576  cu.  in. 

7.  I  tb.  Troy  —  |  oz.  =  ? 

8.  I  bu.  -I-  1  qt.  =  ? 

9.  jfg.  gall.  —  f  qt.  z=  ? 

10.  /^  m.  —  3  rd.  =  ? 

11.  /jsq.m.  +  ||A.  =  ? 

*  Art.  195,  Note. 


132  COMPOUND  DENOMINATE  NUMBERS. 

208.     Multiplication. 
III.  Ex.  1  pt.  3  gi.  X  7  =  what  ? 

Operation.  7  X  3  gi.  =:  21  gi.z=  5pt.-}-l  gi- ;  we  write  1  gi.  under 

1         %'       S^-  ^"  ^^®  multiplicand,  and  reserve  the  5  pt.  to  add 

with  the  pints  in  the  product.    7  X  1  pt.  =:  7  pt.,  which 

with  the  5  pt.  reserved  =  12  pt.  ==:  6  qt.  0  pt.     Ans. 


6  qt.  0  pt.  1  gi  Q  ^^^  Q  p^^  J  gj^     jj^^^^  ^^^ 

Rule  for  Multiplication  of  Compound  Numbers.  Mul- 
tiple/ the  number  of  the  lowest  denomination  hy  the  multiplier,  and 
divide  the  product  thus  obtained  hy  the  number  it  takes  of  that  de- 
nomination to  make  one  of  the  next  higher  ;  write  the  remainder 
under  the  term  multiplied,  and  add  the  quotient  with  the  product 
of  the  next  higher  denomination ;  and  thus  continue  till  all  the 
terms  of  the  multiplicand  are  multiplied. 

Examples. 

1.  7  bu.  4  pk.  3  qt.  1  pt.  X  5  =  ? 

Ans.  40  bu.  2  pk.  1  qt.  1  pt. 

2.  28  gall.  2  qt.  1  pt.  3  gi.  X  17  =  .^ 

Ans.  488  gal.  0  qt.  1  pt.  3  gl 

3.  20  T.  16  cwt.  4  lb.  15  oz.  X  25  z=:  ? 

4.  8  ib,  1  i,  6.3,  2  a  18  gr.  X  37  =  ? 

5.  24  lb.  8  oz.  16  pwt.  5  gr.  X  78  ==  ? 

6.  118°  24 '52"  X  55  =  ? 

7.  18  y.  37  d.  23  h.  14  min.  7  sec.  X  12  =  ?* 

8.  36  y.  48  w.  5  d.  7  h.  3  m.  10  s.  X  21  =  ?♦ 

9.  814  m.  21  ch.  45  1.  X  83  =  ? 

10.  7  £.  6  s.  3  d.  2  qr.  X  73  =:  ? 

11.  4  m.  7  f.  35  rd.  3  yd.  2  ft.  11  in.  X  29  =  ? 

12.  118  cd.  4  cd.  ft.  1  cu.  ft.  X  35  =  ? 

13.  76  cu.  yd.  12  cu.  ft.  184  cu.  in.  X  24  =:  ? 

14.  4  sq.  m.  320  A.  3  R.  25  sq.  rd.  25  sq.  yd.  7  sq.  ft.  117  sq. 
in.  X  9  =  ? 

15.  How  much  land  in  9  gardens  each  containing  34  sq.  rd.  4  sq. 
yd^  8  sq.  ft.  67  sq.  in.  ? 

*  Art.  195,  Note. 


DIVISION.  133 

16,  How  many  pickles  in  17  jars,  each  jar  holding  2  gall.  3  qt. 
1  pt.  ? 

17.  If  a  car  runs  18  m.  3  f.  29  r.  2 J-  y.  in  J-  of  fia  hour,  how 
far  will  it  run  in  7  h.  ? 

1^  For  Dictation  Exercises,  see  Key. 

209.     Division. 
III.  Ex.     17  cwt.  23  lb.  7  oz.  ^  8  =  what? 

Operation.  ^  of  17  cwt.  =  2  cwt.  with  a  remainder 

cwt.     lb.     oz.  of  1  cwt,  which  reduced  to  lbs.  and  added 

8  )  17     23      7  to  the  23  lb.  =  123  lb. ;    l  of  123  lb.= 

lo  lb.  with  a  remainder  of  3  lb.,  which  re- 


2     1 


^'  '^      *         duced  to  oz.  and  added  to  the  7  oz.  ::=:  55 
oz. ;  ^  of  55  oz.  =  6|  oz.     Ans.  2  cwt.  15  lb.  6|  oz.     Hence  the 

Rule  p^or  Division  of  Compound  Numbers.  Divide  the 
highest  term  of  the  dividend  hy  the  divisor  ;  write  down  the  quo- 
tient,  and  reduce  the  remainder  to  its  value  in  the  next  lower 
denomination  ;  add  it  to  the  number  of  that  denomination,  divide 
as  before,  and  thus  continue  till  every  term  is  divided. 

Proof.  Compound  Division  may  be  proved  by  Compound 
Multiplication,  and  Compound  Multiplicaiion  by  Compound 
Division. 

Examples. 

1.  14  £.  11  s.  3  d.  2  far.  -f-  8  =r  ?  Ans.  1  £.  IG  s.  4  d.  3^  far. 

2.  56  cd.  f)  cd.  ft.  14  cu.  ft.  -^  5  =:.  ? 

Ans.  11  cd.  2  cd.  ft.  12  cu.  ft.  691^  in. 

3.  36°  18'  36''  -^  40  =  ? 

4.  74  ch.  3  rd.  22  1.  -^  18  =  ? 

5.  113  cu.  yd.  22  cu.  ft.  Ill  cu.  in.  -^  42  =.? 

6.  16  lb.  5  oz.  3  pwt.  21  gr.  -i-  13  =  ? 

7.  38  lb,  8  i,  7  5,  2  9,  6  gr.  -f-  15  =? 

8.  27  T.  14  cwt.  2  qr.  16  lb.  12  oz.  3  dr.  ~  6  z=:  ? 

9.  115  bu.  3  pk.  2  qt.  1  pt.  -^  19  =? 

10.  136  gall.  3  qt.  1  pt.  -f-  41  —  ? 

11.  1  m.  5  f.  37  r.  2  yd.  2  ft.  9  in.  -f-  12  =:  ? 

12.  365  A.  3  R.  19  sq.  rd.  28  sq.  yd.  4  sq.  ft.  110  sq.  in.  -^ 
71  =? 


134  COMPOUND  DENOMINATE  NUMBERS. 

13.  30  y.  35  d.  7  h.  20  min.  35  sec.  —  29  ziz  ? 

14.  How  far  must  a  bird  fly  in  one  minute  to  fly  55  mile*  in  an 
hour? 

15.  If  37  bu.  4  pk.  of  rye  be  divided  between  7  men,  what 
will  each  man  receive  ? 

IG.  If  65  A.  2  R.  18  sq.  rd.  10  sq.  yd.  1}  sq.  ft.  be  divided 
into  55  house-lots,  what  is  the  size  of  each  ? 

17.  How  long  will  it  take  to  travel  1  mile,  at  the  rate  of  75 
miles  in  10  h.  18  min.  12  s.  ? 

18.  Among  how  many  men  may  624  gall.  3  qt.  be  divided, 
that  each  man  may  receive  12  gall.  3  qt.  ? 

Note.  —  Reduce  each  of  the  above  to  quarts  before  dividing. 

19.  How  many  bins,  each  containing  5  bu.  3  pk.,  will  be  re- 
quired to  hold  885  bu.  2  pk.  of  potatoes  ? 

20.  If  a  man  walks  3  m.  6  fur.  26  rd.  in  one  hour,  how  long 
will  it  take  him  to  walk  23  m.  7  fur.  9  rd.  ? 

1^^  For  Dictation  Exercises,  see  Key. 

310*  Longitude  and  Time. 
As  the  earth  turns  upon  its  axis  once  in  24  hours,  it  follows 
that  2V  ^^  360°,  or  15°  of  longitude,  must  pass  under  the  sun  in  1 
hour,  and  ^^  of  15°,  or  15',  must  pass  under  the  sun  in  1  min.  of 
time,  and  g^\y  of  15',  or  15,"  must  pass  under  the  sun  in  1  sec.  of 
time  ;  or,  in  a 

Tabular  Form. 
15°  of  longitude  make  a  difference  of  1  hour  in  time. 
15'  "  "  "         1  minute  in  time. 

15"  "  "  «         1  second  in  time. 

Hence,  to  find  the  difference  of  longitude  between  any  two 
places :  Multiply  the  difference  of  time  between  the  two  places,  ex- 
pressed  in  hours,  minutes  and  seconds,  by  15.  The  product  will 
express  the  number  of  degrees,  minutes  and  seconds  required. 

Note.  —  As  the  earth  turns  from  west  to  east,  sunrise  occurs  earlier  in 
places  east  and  later  in  places  west  of  any  given  point.  Hentee  the  time 
is  later  in  all  places  east,  and  earlier  in  all  places  west,  of  any  given  pomt 
than  it  is  at  that  point. 


REVIEW.  135 

Examples. 
Note  -^Fot  table  of  latitude  and  longitude  of  places,  see  Art.  206, 
page  13C. 

1.  The  time  in  Pittsburg  is  35  m.  54  s.  earlier  than  in  Boston; 
what  is  the  difference  of  longitude  between  the  two  places  ? 

Ans.  8°  58'  30''. 

2.  What  is  the  longitude  of  Pittsburg  ?  Ans.  80°  2'  W. 

3.  The  time  at  St.  Paul's  is  1  h.  16  m.  19|  s.  earlier  than  in 
New  York  ;  what  is  the  longitude  at  St.  Paul's  ? 

4.  The  time  in  Copenhagen  is  50  m.  19|  s.  later  than  in 
Greenwich  ;  what  is  its  longitude  ? 

5.  The  time  in  Naples  is  5  h.  41  m.  14^  s.  later  than  in  Boston ; 
what  is  its  longitude  ?  Aiis.  14°  15'  3"  E. 

311.     From  Art.  210  we  also  derive  the  following 
Rule.    To  find  the  difference  of  time  between  any  two  places : 
Divide  the  difference  in  longitude,  expressed  in  degrees,  minutes 
and  seconds,  hy  15.      The  quotient  will  he  the  number  of  hoursy 
minutes  arid  seconds  required. 

1.  What  is  the  difference  of  time  between  Albany  and  Boston  ? 

Ans.  10  m.  44f  s. 

2.  Between  Paris  and  St.  Petersburg? 

3.  Between  Montreal  and  Mexico  ? 

4.  Between  Cape  Horn  and  Cape  Good  Hope? 

Ans.bh.  43  m.  Oj\s. 

5.  Between  Charleston,  S.  C,  and  Calcutta  ? 

6.  Between  Canton  and  San  Francisco  ? 

7.  When  it  is  8  o'clock  P.  M.  in  Washington,  what  is  the 
time  in  London  ?  Ans.  1  o'clock,  7  m.  37^  s.  A.  M.  of  the 
next  day. 

8.  At  2  A.  M.,  Jan.  1,  1864,  at  Paris,  what  was  the  time  in 
New  Orleans? 

313.     Questions  for  Review. 
1.   When  are  denominate  numbers  simple  ?  when  compound  ? 
Give  examples  of  each.     May  abstract  numbers  be  compound  ? 
What  is  Reduction  as  applied  to  compound  numbers  ?    What  is 


130                    COMPOUND  DENOMINATE  NUMBERS. 
REDUCTION-  DESCENDING? REDUCTION  ASCENDING?    Rule 

reduction  descending.  Write,  perform,  and  explain  an  example, 
illustrating  it.  Rule  for  reduction  ascending.  Illustrate  it.  How 
can  you  prove  examples  in  reduction  descending  ?  in  reduction 
ascending  ? 

2.  What  are  the  denominations  in  United  States  Money? 
the  coins?  IIow  are  the  gold  coins  hardened?  the  silver?  Give 
the  table  of  U.  S.  or  Federal  Money.  From  what  are  the  names 
of  the  denominations  derived  ?  Which  place  from  the  decimal 
point  do  the  mills  occupy  ?  How  will  you  write  forty-five  thou- 
sand three  hundred  twenty-five  mills?  How  many  dollars  in 
the  above  number  of  mills  ?  how  many  cents  ? 

Give  the  table  of  English  Money.  Name  the  denomina- 
tions. What  is  a  guinea  ?  a  crown  ?  a  sovereign  ?  value  of  a 
pound  English  money  in  Federal  money  ? 

3.  Name  the  tables  of  Weight.  Repeat  the  table  of  Troy 
Weight ;  of  Apothecaries'  Weight ;  of  Avoirdupois  Weight.  Which 
is  in  most  common  use  ?  By  which  would  you  buy  and  sell  coal  ? 
iron  ?  silver  ?  salt  ?  quinine  ?  fish  ?  emeralds  ?  flour  ?  gold  ? 
opium  ?  What  is  a  long  ton  ?  Which  is  the  most,  a  lb.  Avoirdu- 
pois or  a  lb.  Troy  or  Apothecaries'  Weight  ?  an  oz.  Avoirdupois, 
or  an  oz.  Troy  or  Apothecaries'  Weight  ?  1  lb.  Avoirdupois  =: 
how  many  grs.  Troy  or  Apothecaries'  Weight  ? 

4.  Name  the  table  of  Extension  in  Length.  Repeat  the 
table  of  Long  Measure.  Draw  a  line  that  you  think  to  be  1  inch 
Ipng.  Divide  it  into  lines.  Mark  off  one  foot  on  your  slate  or  paper. 
Measure  the  distance  from  your  home  to  the  school-house.  What 
is  a  land  league  ?  How  many  English  miles  =  1°  on  the  earth's 
surface?  how  many  geographical?  How  is  cloth  usually  meas- 
ured? Give  the  denominations  of  Surveyors'  Measure.  Repeat 
the  table.  Denominations  of  Mariners'  Measure  ;  —  the  table  ? 
Which  is  longer,  a  land  or  sea  league  ? 

5.  What  are  the  denominations  of  Square  Measure  ?  Repeat 
the  table.  Draw  a  square  1  inch  each  way  ;  —  ^  inch  each  way. 
What  part  of  the  first  square  is  the  second  ?  Difference  between 
5  square  inches  and  5  inches  square  ?     Define  a  rectangle ;  a 


REVIEW.  137 

square.  Show  why  you  multiply  the  length  of  a  rectangle  by 
the  breadth  to  obtain  the  surface.  Can  you  multiply  feet  by 
yards  ?  When  the  length  of  one  side  of  a  rectangle  is  given  in 
feet,  and  the  other  in  rods,  how  do  you  find  the  surface  ?  When 
the  square  contents  and  one  dimension  are  given,  how  do  you 
find  the  other  ?     Find  the  area  of  the  top  of  your  desk. 

6.  Give  the  denominations  of  Cubic  Measure  ; — the  table. 
Give  the  table  for  Wood  Measure.  Define  a  parallelopiped.  How 
do  you  find  its  contents  ?  Illustrate.  When  the  solid  contents 
and  two  dimensions  are  given,  how  do  you  find  the  third  ?  How 
many  faces  has  a  cube  ?  How  many  edges  ?  How  many  cubic 
feet  of  air  would  your  school-room  contain  if  there  were  nothing 
else  in  it?  Suppose  the  average  number  of  pupils  who  attend 
your  school  were  shut  up  in  the  school-room  without  any  means  of 
ventilation,  how  long  before  they  would  all  die,  if  each  person 
should  render  20    cubic  feet  of  air  per  hour  unfit  to  sustain  life  ? 

7.  Repeat  the  denominations  of  Liquid  Measure  ;  —  the  ta- 
ble.    What  is  the  common  size  of  barrels  and  hogsheads  ? 

8.  Eepeat  the  denominations  of  Dry  Measure  ;  —  the  table. 
Which  is  the  larger,  1  quart  Liquid  or  1  quart  Dry  Measure  ? 
1  bushel  Dry  Measure  =  how  many  gallons  Liquid  Measure  ? 

9.  Where  is  Circular  Measure  used  ?  Define  circle ;  circum- 
ference; arc;  radius;  diameter;  degree ;  minute  ;  second ;  semi- 
circumference  ;    quadrant  ;    sextant  ;    sign.      Give    the    table. 

d 

Define  an  angle;    the  vertex.      Read  the  annexed  angle,  e<^. 

How  is  an  angle  measured  ?  Does  the  size  of  an  angle  depend 
at  all  upon  the  length  of  its  sides  ?  What  is  a  right  angle  ?  How 
are  its  sides  in  regard  to  each  other  ?  What  is  an  obtuse  angle  ? 
an  acute  angle  ?  How  many  right  angles  can  you  have  at  the 
centre  of  a  circle  ?  Dmw  a  right  angle ;  an  obtuse  angle ;  an 
acute  angle ;  a  circle  ;  a  semi-circle ;  an  arc ;  a  radius  ;  a  diam- 
eter ;  a  sextant ;  a  quadrant. 

10.  What  is  an  astronomical  day  ?  a  solar  day  ?  Which  is  the 
longer  ?  How  is  the  solar  day  divided  ?  Give  the  denomina- 
tions of  time*     Repeat  the  Table.     Give  the  reason  for  leap 


138  COMPOUND  DENOMINATE  NUMBERS. 

year ;  the  rule.  What  are  the  seasons  of  the  year,  and  how 
divided  ?  Give  the  number  of  days  in  each  calendar  month. 
Repeat  the  helping  lines.  Repeat  the  table  of  numbers  under 
the  head  of  Miscellaneous  ;  of  Paper ;  of  Books.  What  measure? 
are  sometimes  used  for  animals  ? 

11.  How  do  you   reduce  a  fraction  of  one  denomination  to 
whole  numbers  of  lower  denominations?     How  do  you  rec^*' 
whole  numbers  of  lower  denominations  to  the  fraction  of  a  higher? 

12.  What  is  Compound  Addition;  Subtraction;  Multi- 
plication ;  Division  ?  How  do  these  operations  differ  from 
similar  operations  upon  simple  numbers  ?  Give  an  example  in 
each,  and  repeat  the  rule.    How  can  you  prove  these  operations  ? 

13.  How  do  you  find  the  difference  of  latitude  between  two 
places  upon  the  same  side  of  the  equator  ?  upon  different  sides  ? 
Give,  in  your  own  words,  a  rule  for  finding  the  difference  of 
longitude  between  any  two  places.  Suppose  two  places  are  in 
different  longitudes,  and  in  your  operation  the  number  of  degrees 
between  them  is  found  to  exceed  180°,  what  will  you  do  ? 

14.  How  do  you  add  or  subtract  denominate  fractions  f 

15.  When  the  difference  of  time  between  any  two  places  i? 
given,  how  do  you  find  the  difference  of  longitude  f  When  the 
difference  of  longitude  is  given,  how  do  you  find  the  difference 
of  time  ?  For  places  east  of  any  given  point,  must  the  difference 
of  time  be  added  or  subtracted  to  give  the  true  time  ?  For  places 
west,  what  must  be  done  ? 

S13«    Miscellaneous  Examples. 

1.  If  coal  is  worth  $9^  a  ton,  what  is  the  expense  of  a  coal 
fire  for  a  week,  allowing  it  consumes  25  lbs.  a  day,  coal  being 
sold  by  the  long  ton  ? 

2.  What  will  be  the  cost  of  freighting'50  bbls.  of  flour,  at  $.382 
per  bbl.  ? 

3.  A  quantity  of  gold  weighed  4  lb.  10  oz.  3  pwt.  before  re- 
fining, and  3  lb.  11  oz.  2  pwt.  9  gr.  afterwards.  What  was  lost 
in  the  process  ? 

4.  If  £69  12s.  be  paid  for  6  cwt.  of  tobacco,  what  is  the  price 
per  pound  ? 


MISCELLANEOUS  EXAMPLES.  13d 

5.  An  apothecary  mixed  3  tb,  10  §,  2  3,  2  9,  14  gr. 

lib,    4i,  13,  2  9,  17gr. 

2tb,    7i,  6  3,  19,  13gr. 
and  divided  the  mixture  into  100  equal  parts.     What  was  the 
weight  of  each  part  ? 

6.  What  is  the  duty  on  6  lb.  4  oz.  of  essence  of  lemon,  at  $.50 
per  pound  ? 

7.  What  is  the  cost  of  137  gall.  2  qts.  of  molasses  at  12^  cts. 
per  quart? 

8.  If  a  bird  fly  1°  in  1  h.  8  m.  15  s.,  in  what  time  will  it  fly 
round  the  world  at  the  same  rate  ? 

9.  How  many  times  will  a  wheel  3  ft.  4  in.  in  circumference 
turn  in  crossing  a  bridge  that  is  40  rd.  1  yd.  2  ft.  long  ? 

lOr  What  will  be  the  cost  of  125  pieces  of  delaine,  averaging 
33  yards  in  length  and  22  inches  in  width,  at  25  cts.  per  yard  in 
length,  and  2  cts.  per  sq.  yard  for  duties  ?  Ans,  $1081. 66§. 

11.  How  many  bushels  will  a  bin  contain  which  is  10  ft.  long, 
8  ft.  wide,  and  5  ft.  deep  ? 

12.  At  $6.00  a  cord,  what  cost  a  pile  of  wood  33  ft.  long,  8  ft. 
10  in.  high,  and  4  ft.  wide  ? 

13.  Divide  an  arc  of  15°  12'  3"  by  7^. 

14.  Reduce  f  of  a  great  gross  to  integers  of  lower  denomina. 
tions. 

15.  What  will  be  the  cost  of  fencing  a  lot  of  land  20  rods  by 
260  rods,  at  12^  cts.  per  foot  ? 

16.  A  farmer  divided  one  half  of  his  estate  of  350  A.  3  R. 
20  rd.  equally  between  his  two  daughters,  and  the  balance,  after 
setting  off  17j-  A.,  equally  between  his  tWo  sons.  What  was 
the  share  of  each  son  and  daughter  ? 

17.  How  many  cords  of  wood  in  25  loads,  each  measuring 
1  cd.  1  cd.  ft.  12  cu.  ft.? 

18.  What  would  be  the  cost  of  the  above  at  $4.50  per  cord  ? 

19.  How  many  yards  of  carpeting  1^  yd.  wide  will  cover  a 
floor  18  ft.  sq.? 

20.  If  a  cotton  mill  can  make  1200  yds.  of  cloth  per  hour,  how 
many  yards  could  be  made  by  working  10  hours  a  day  from  July 
7th  to  January  4th,  allowing  for  26  Sabbaths  ? 


140  COMPOUND  DENOMINATE  NUMBERS. 

21.  Charge  15  lb.  8  oz.  A  v.  to  pounds  and  ounces  Troy. 

22*  How  many  cakes  of  ice  1^  ft.  sq.  by  1  ft.  thick  may  be 
contained  in  a  building  measuring  in  the  inside  105  ft.  long,  60  ft. 
wide,  31  ft.  3  in.  high  ? 

23.  How  many  bricks  8  in.  by  4  in.  will  cover  a  court  75  ft. 
by  50  ft.  ? 

24.  How  many  sq.  ft.  does  the  surface  of  a  box  contain,  which 
is  3  ft.  long,  2  ft-  wide,  and  6  ft.  deep  ? 

25.  What  is  the  price  of  250  tons  of  lead  at  $0.11  per  pound? 

26.  I  have  imported  1  T.  5  cwt.  1  qr.  15  lbs.  of  black  lead, 
which  cost  me  at  New  York  $200.00  a  ton,  and  on  which  I  have 
also  paid  $10.00  a  ton  for  duties  ;  for  what  must  I  sell  it  per 
pound  to  gain  $150.00  on  the  lot,  if  I  buy  and  import  by  the  long 
ton?         '  Ans.$.imm. 

27.  At  $6.50  for  two  dozen  pints  of  olive  oil,  what  cost  1  qt.  ? 
28!  What  is  the  diiference  between  37  f.  8  rd.  0  yd.  1  ft.  3  in., 

and  37  f.  7  rd.  5  yd.  2  ft.  9  in.  ? 

29.  I  have  sold  7^  tons  of  chalk  for  $75.30.  What  do  I  re- 
ceive per  pound  ? 

30r  A  regiment  of  troops  that  enlisted  for  9  months  was  not 
discharged  till  July  20th,  1863,  which  was  1  mo.  26  d.  after  the 
term  of  service  had  expired.     When  did  they  enlist  ? 

31.  Divide  60  miles  by  7,  carrying  out  the  quotient  to  the  low- 
est denomination. 

32.  From  a  pile  of  wood  48  ft.  long,  4  ft.  high,  and  4  ft.  wide, 
was  sold  at  one  time  3  cd.  5  cd.  ft. ;  at  another  2  cd.  32  cu.  ft. ; 
what  is  the  remainder  worth  at  $4  per  cord  ? 

33.  What  is  the  talue  of  7  cd.  7  cd.  ft.  and  5  cd.  112  cu.  ft.  of 
wood,  at  S7  per  cord  ? 

34.  If  a  man  saves  1  hour  50  minutes  a  day  by  habits  of  order, 
1  hour  30  minutes  by  promptness  in  business,  and  half  an  hour 
by  early  rising,  how  much  time  is  saved  in  25  years  of  365| 
days  each? 

35.  A  floor  30  ft.  by  12  ft.  is  to  be  covered  with  carpeting  | 
of  a  yard  wide.     Required  the  number  of  yards. 

36.  Bought  7^  tons  of  coal  for  $75.30  ;  what  was  the  cost  per 
cwt.? 


MISCELLANEOUS  EXAMPLES.  141 

37.*  A  single  block  of  quartz  in  Australia  is  said  to  have  yielded 
$32000  worth  of  gold.  At  $16  an  oz.  Troy,  what  was  the  weight 
of  the  metal  in  Avoirdupois  ? 

38.  Bought  5  oz.  7  pwt.  1 2  gr.  of  gold-foil  at  $30  per  ounce ; 
3  oz.  15  pwt.  of  gold-plate  at  $18.75  per  ounce;  what  was  the 
amount  of  my  bill  ? 

39.  How  many  cubic  feet  in  the  hold  of  a  vessel  which  con- 
tains 2000  bushels  of  grain  ? 

40.  Southampton  is  in  longitude  I''  30'  W.  New  York  is 
about  74°  W.  Would  a  passenger  on  arriving  at  New  York  from 
Southampton  find  his  watch  too  fast  or  too  slow,  and  by  how 
much,  if  right  for  Southampton  time  ? 

41.  Two  vessels  are  100°  apart,  and  sailing  toward  each  other ; 
one  sails  2°  50'  2"  in  a  day,  and  the  other  3°  10'  45"  in  the  same 
time.     How  far  apart  will  they  be  at  the  end  of  10  days? 

42.  A  steam  frigate,  sailing  at  the  rate  of  15§  miles  an  hour, 
gives  chase  to  a  pirate  vessel,  5^  miles  ahead,  sailing  at  the  rate 
of  14^  miles  an  hour;  in  what  time  will  the  frigate  overtake  the 
pirate?  Ans,  4^  h. 

43.  How  many  feet  of  lumber  in  a  piece  of  square  timber  10 
inches  wide,  6  inches  thick,  and  9  feet  long  ?  Ans.  45  ft. 

Note.  —  Lumbor  is  considered  1  inch  in  thickness. 

44.  In  50000  feet  of  lumber  how  many  cords,  cord  feet,  and 
cubic  feet  ?  Ans.  32  cd.  4  cd.ft.6f  cu.  ft. 

45.  How  many  cords  and  cord  feet  of  wood  can  be  put  into  a 
shed  8  ft.  by  18  ft.  7  in.,  and  10  ft.  5  in.  high  ? 

46.  How  many  cords,  and  what  will  be  the  cost  at  $4.56  per 
cord,  of  wood  in  a  pile  404^  ft.  long,  6  ft.  high,  and  8  ft.  wide? 

47.  A  man  purchased  75  cords  of  wood  for  $360 ;  he  sold  the 
following  lots,  10  cd.-64  cu.  ft.,  15  cd.  80  cu.  ft.,  and  llf  cd.,  all 
at  $5  per  cord ;  what  did  he  gain  on  what  he  sold  ? 

48.  What  would  be  the  cost  of  sawing  the  remainder  of  the 
75  cords,  if  it  is  worth  25  cents  to  saw  2  cd.  ft.? 

49.  How  many  barrels  of  31  gallons  each  will  be  contained  in 
a  water  tank  3  ft.  square  and  4  ft.  3  in.  deep  ? 


142  COMPOUND  DENOMINATE  NUMBERS. 

50.  What  was  the  cost  per  pound  for  lead,  5  lbs.  to  the  sq.  ft., 
to  line  the  above  tank,  if  the  whole  cost  $38f  ? 
Sit  What  part  of  1  m.  5  f.  is  2  m.  7  f.  ? 

52.  What  is  the  area  of  a  lot  of  land  25  chains  long  and  17 
rods  wide  ? 

53.  How  many  cubic  inches  in  2  bu.  1^  pk.  2^  qt.  dry  meas- 
ure, and  5  gall,  liquid  measure  ? 

54.  What  is  the  bill  for  ^  dozen  silver  spoons,  each  M'eighing 
2  oz.  9  pwt.  12  gr.  at  $1.50  per  ounce,  and  2  T.  o  cwt.  2  qr.  of 
iron  at  $3  per  cwt.  ? 

55.  When  it  is  noon  at  London,  what  is  the  time  in  Lawrence, 
Mass.,  71°  20'  W.  ? 

56^  What  will  it  cost  to  paper  a  room  16  ft.  6  in.  by  14  ft.,  and 
7  ft,  high,  with  paper  |  yds.  wide,  8  yds.  in  a  roll,  $.75  a  roll,  bor- 
dering at  the  top  of  the  wall  being  3  cents  a  yard,  and  overlap- 
ping the  paper  by  the  width  of  the  border,  no  allowance  being 
made  for  windows  and  doors  ?  Ans.  $7.72§. 

57^  How  many  rolls  of  paper,  20  inches  wide  and  8  yards  long, 
will  paper  the  walls  of  a  room  18  ft.  by  16  ft.  and  10  ft.  high,  in 
which  are  two  doors,  each  6  ft.  by  2J^  ft.,  and  4  windows,  each 
6  ft.  by  21- ft.? 

58!  How  many  cubic  yards  of  earth  must  be  removed  to  dig  a 
ditch  3  ft.  wide  and  2^  ft.  deep  outside  of  a  lot  of  land  40  rods  by 
38  rods,  lOi  ft.  Ans.  724^  yds. 

Note.  —  In  order  that  the  ditch  may  entirely  surround  the  land,  twice 
its  width,  or  6  ft.,  must  be  added  to  either  the  length  or  width  of  the 
land.  Adding  it  to  the  width,  we  have  for  the  entire  length  of  the  ditch 
2  X  40  rd.  +  2  X  39  rdj.  =  158  rd.  The  pupil  will  see  this  more  clearly  by 
making  a  drawing  of  the  land  and  ditch. 

59*  How  many  bricks  in  the  walls  of  a  building  29  ft.  long  by 
24  ft.  wide  and  30  ft.  high,  the  walls  being  2  ft.  thick,  and  the 
bricks  8  in.  by  4  in.  by  2  in.  ?  Ans,  158,760  bricks. 

60.  How  much  carpeting  |  yd.  wide  will  cover  a  block  3  ft. 
long,  8  inches  wide,  and  6  inches  high  ? 

61.  Add  ?  of  the  month  of  February,  1860,  to  \  of  the  days 
from  February  25,  1861,  to  May  6,  1861. 


MISCELLANEOUS   EXAMPLES.  143- 

62!  Which  will  cost  more,  and  how  much  more,  15  times  2 
;wt.  24  lbs.  of  lead  at  2£  4d.  a  cwt.,  or  15  lbs.  1|  oz.  silver  at 
)s.  9d.  per  ounce  ? 

G3.  Suppose  a  boat  to  be  moved  forward  through  a  strait  1 
oaile  in  length,  by  steam  100  ft.  a  minute,  by  sail  25  ft.  a  minute, 
md  by  the  current  30  ft.  a  minute  ;  how  long  will  it  be  in  going 
the  length  of  the  strait  ? 

G4.  How  long,  if  it  were  moving  in  the  opposite  direction,  pro- 
pelled  only  by  steam  ? 

65.*  What  is  the  difference  between  ^  of  $10.50  and  ^  of  |^  of 
7£  6s.  lOd.?     (Ans.  in  $.) 

66^  The  Fitchburg  railroad,  67  m.  6  f.  24  rd.  o^qq  yd.  long, 
was  built  for  $3540000  ;  what  was  the  cost  per  mile  ? 

67!  How  many  ])aving-stones  6  in.  by  8  in.  will  be  required  to 
pave  a  street  27  rods  long  by  50  ft.  wide  ? 

68!  A  druggist  bought  8  lbs.  Dover's  powder  at  $2  per  lb.  Av., 
and  sold  it  in  separate  i)owders,  7  grs.  to  a  powder,  at  the  rate  of 
4  for  6  cts. ;  what  did  he  gain  ?  Ans.  $104. 

69!  An  apothecary  mixed  5  §,  1  5,  2  9  of  aloes,  for  which  he 
paid  $1  a  pound,  with  7  §,  6  5,  1  9,  12  gr.  of  rhubarb,  for  which 
}ie  paid  $4  a  pound,  and  made  of  the  mixture  into  pills,  which  he 
sold  in  boxes,  75  grains  in  each  box,  for  25  cents  a  box ;  what 
does  he  gain?  A?is.  $17.802^«j. 

.  70!  In  how  many  days  will  a  locomotive,  which  makes  two 
trips  from  Boston  to  Providence  daily  (the  distance  from  B.  to 
P.  being  61  m.  6  f.  16  r.),  run  5592  m.  1  f.  4  r.  ? 

7lt  How  many  cords  of  wood  can  be  put  into  a  building  meas- 
uring on  the  outside  40  ft.  by  31  ft.  and  15  ft.  high,  the  walls 
being  6  in.  thick  ? 

72!  What  will  be  the  cost,  at  18^  cents  per  cubic  yard,  for  re- 
moving the  earth  to  build  a  cellar  12  feet  deep  whose  measure- 
ment inside  of  the  wall,  which  is  3  ft.  4  in.  thick,  is  27  ft.  long  hj 
15  ft.  wide?  .    .     " 

73.  What  is  the  average  width  of  a  board  whose  edges  are 
straight,  the  width  being  1  ft.  7  in.  at  one  end,  and  1  ft.  9  in.  at 
the  other  ? 


144  COMPOUND  DENOMINATE  NUMBERS. 

74.  What  is  the  length  of  a  board  1  ft.  8  in.  wide,  which  con 
tains  38 1  sq.  ft.  ? 

75.  Estimate  the  cost  of  feeding  a  pair  of  oxen  through  the 
winter  of  1863  and  1864,  if  1  ox  weighed  1772  lbs.  and  the  other 
1431  lbs.,  and  hay  was  $13.75  per  ton,  and  the  oxen  were  al- 
lowed -^jj  of  their  weight  in  hay  each  day. 

76.  What  is  the  length  of  a  stick  of  timber  which  is  17  inches 
square,  and  contains  154  ft.  120  in.  cu.  measure? 

77.  If  a  druggist  sells  1  gross  2  doz.  papers  of  bitters  a  day, 
how  many  will  he  sell  from  the  19th  of  Dec,  1859,  to  15th  Mar., 
1860,  deducting  12  Sundays? 

78.  A  man  sold  a  sheep  for  1^£,  a  calf  for  |£,  and  a  fowl 
for  f  s.  |d.  ;  what  did  he  receive  for  them  all  ? 

79t  What  is  my  tax  on  silver,  consisting  of  a  lot  of  spoons 
weighing  3  lbs.  ID  oz.  9  dr.  Avoirdupois ;  1  doz.  forks  weighing 
2  lbs.  8  oz.  5  dr.,  basket  and  other  articles  of  plate  weighing 
2  lbs.  5  oz.  2  dr.,  at  3  cts.  per  oz.  Troy,  40  oz.  Troy  being  exempt  ? 

A71S.  $2.51 1 . 

80^  What  will  be  the  cost  of  bricks,  at  $7  per  M.,  to  construct 
the  walls  of  a  building  100  ft.  by  40  ft.,  36  ft.  high,  the  walls 
being  15  in.  thick,  in  Avhich  are  two  doors  each  7  ft.  by  3  ft.  and 
24  windows  3  ft.  3  in.  by  6  ft.  6  in.,  the  bricks  being  of  the 
usual  size,  and  no  allowance  made  for  mortar  ? 

8 it  What  will  the  bricks  cost  to  construct  the  walls  and  bot- 
tom of  a  cistern  whose  inside  dimensions  are  8  ft.  by  8  ft.  and 
6  ft.  deep,  the  walls  and  bottom  to  be  1  ft.  thick  ? 

82r  When  snow  is .  uniformly  6  inches  deep,  how  many  cubic 
feet  are  there  on  one  acre  of  land  ? 

83!  What  must  be  the  depth  of  a  ditch  around  a  garden  out- 
side, 5  rods  by  2  rods,  the  ditch  1  ft.  wide,  that  the  earth  taken 
from  the  ditch  may  raise  the  surface  2  inches  ?  Ans.  1  ft.  11/^  in. 

84t  How  many  square  feet  in  a  walk  around  a  garden  inside 
and  next  to  the  fence,  the  garden  being  27^  rods  long,  20^  rods 
wide,  the  walk  being  4  feet  wide  ?  Ans.  6239  ft. 

85.  When  it  is  7  o'clock  P.  M.  at  Boston,  what  is  the  time  at 
St.  Petersburg  ? 


DUODECIMALS,  145 

86!  Suppose  a  vessel  to  go  up  stream  by  the  power  of  steam 
at  the  rate  of  16  miles  an  hour,  by  sail  at  the  rate  of  4  miles, 
and  to  be  set  back  by  the  current  at  the  rate  of  2  miles  an  hour ; 
in  what  time  will  another,  which  is  propelled  forward  at  the  rate 
of  25  miles  an  hour  and  is  set  back  the  same  as  the  former,  over- 
take her,  if  she  starts  3  hours  later  ? 


A   DUODECIMALS* 

dl4llr*  Duodecimal  Fractions,  or  Duodecimals,  are  frac- 
tions whose  denominators  are  12  or  some  integral  power  of  12. 
They  may  also  be  considered  as  a  kind  of  compound  numbers, 
the  values  of  whose  denominations  vary  by  a  uniform  scale  of  12. 

Duodecimals  are  sometimes  used  in  computing  lengths,  sur- 
faces, and  solids ;  but  all  examples  in  mensuration  can  be  per- 
formed by  the  use  of  common  or  decimal  fractions. 

^\5o  The  denominations  are  feet,  primes  ('),  seconds  ("), 
thirds  ('"),  fourths  ("'')»  ^^hs  (^),  &c.  The  foot  is  considered 
the  unit;  primes  are  12ths  of  feet;  seconds  12ths  of  primes  or 
144ths  of  feet;  thirds  12ths  of  seconds,  144ths  of  primes,  or 
1728ths  of  feet,  &c.  Hence,  in  length,  inches  are  represented 
by  primes,  in  surface  by  seconds,  and  in  solids  by  thirds. 

The  marks  which  indicate  the  degree  of  the  denominations 
are  called  Indices. 

316.     Table. 

1  foot  =12*.  l"/  =  12^''^ 

l'  =  12".  V'"^\2\ 

V'=.\2"',  etc.     etc. 

Units  X  primes,  seconds,  &c.  =:  primes,  seconds,  &c. 
Primes  (^*^s.)  X  primes  (^^s.)  z=:  seconds  (y^s.) 
Primes  (yVO  ^  seconds  (yl^s.)  =  thirds  (i^^VfS.) 
Primes  (y^^s.)  X  thirds  (y-^s,)  zr:  fourths  (^g^fg^s.) 
Seconds  (y^s.)  X  seconds  (y^s.)  r=  fourths  {^wkr^^-)   ^ 
etc.  etc.       ^ 

10  *  Optinal. 


146  COMPOUND  DENOMINATE  NUMBERS. 

SIT.  Duodecimals  may  he  added,  sidjtr acted,  midtipUed,  and 
divided  like  compound  numbers,  it  being  borne  in  mind  that  a  unit 
of  any  denomination  is  12  times  one  of  the  next  lower  denomina- 
tion, and  ^  of  one  of  the  next  higher. 

Illustrative  Examples, 
addition.  subtraction. 

2     3'  8''     4'"  18       2'  4"     3'" 

5     3'  9''     2'"  3     11'  8"     4.'" 

^   ^^    ^     ^^  Ans.    14  ft.  2'  1"  \V" 


Ans,  15  ft.  5'  8''  ^" 

MULTIPLICATION. 
DIVISION.  25       3'       2" 

^  8)365     iV     8''  4'"  8""  7 

Ans,     45  ft.  8'  10"  0'"  1""  Ans.  176  ft.  10'  2" 

Examples. 

1.  What  is  the  sum  of  the  contents  of  3  blocks  of  granite  con- 
taining severally  92  ft.  11'  7"  6'",  484  ft.  1'  9",  472  ft.  6'? 

Ans.  1049  ft.  T  4"  Q'". 

2.  If  from  a  board  measuring  31  ft.  7',  there  be  cut  19  ft.  11' 
4"  9f",  what  will  remain  ?  Ans.  11  ft.  7'  7"  2f". 

3.  Required  the  contents  of  5  blocks  of  marble,  each  contain- 
ing 4  ft.  3'  9".  Ans.  21ft.  6'  9". 

4.  There  being  679  ft.  7'  6"  of  glass  in  29  windows  of  equal 
size,  how  much  glass  does  one  window  contain  ? 

Ans.  23  ft.  5'  2|^". 

5.  There  are  3049  ft.  3'  0"  8'"  of  glazing  in  my  dwelling- 
house  and  two  equal  green-houses.  My  dwelling-house  contains 
679  ft.  7'  6"  9'".  What  is  the  quantity  of  glass  in  each  green- 
house ?  Ans.  1184  ft.  9'  8"  11'"  &"-\ 

S18.  It  only  remains  to  multiply  and  divide  duodecimals  by 
duodecimals.  These  operations  can  be  easily  performed,  if  we 
observe  that  the  index  of  the  product  of  any  two  terms  equals  the 
sum  of  the  indices  of  the  terms  themselves,  and  the  index  of  the 
quotient  of  one  term  divided  by  another^  equals  the  difference  of 


DUODECIMALS.  147 

the  indicei  of  the  dividend  and  divisor.     Thus,  2'  X  4"  =  8^"; 
12""  -f-  3"'  =  4'. 

^19.     Multiplication. 
III.  Ex.  What  is  the  area  of  a  floor  measuring  12  ft.  3'  in 
length  and  10  ft.  6'  in  breadth? 

Operation.  &  X  ^' —  18"  =  1'  +  6".    We  write  the  6"  and 

12ft.  3'  reserve  the  V  to  add  with  the  primes.     6'  X  12  =: 

10ft.  6'  72',  which,  with  the  1'  reserved  =  73'=  6  ft.  -f  1', 

Q      y~Qn  which  we  write  in  their  proper  places.     10  X  3'  = 

J  22      6'  ^^'  =  2  ft.  +  6'.     We  write  the  6'  and  reserve  the 

2  ft.  to  add  with  the  feet.  10  X  12  ft.  =  120  ft. ;  120 

128ft.  r  6"  Arts.  ft.  +  2  ft.  =  122  ft.      Adding  the  partial  products 
we  obtain  for  the  answer  128  ft.  7'  6".     Hence  the 

Rule  for  Multiplication  of  Duodecimals.  Beginning 
with  the  lowest  denomination,  multiply  all  the  terms  of  the  multi- 
plicand  hy  each  term  of  the  multiplier  separately ;  divide  each 
product  by  12  (except  when  the  product  is  feet) ;  write  the  re- 
mainder, and  reserve  the  quotient  to  add  to  the  next  product. 
Give  to  every  term  thus  obtained  an  index  equal  to  the  sum  of 
the  indices  of  its  two  factors.  The  sum  of  the  partial  products 
will  be  the  entire  product. 

Examples. 

1.  Multiply  7  ft.  4'  by  5  ft.  2'.  Ans.  37  ft.  10'  8". 

-1  2.  ;Multiply  4  ft.  8'  5"  by  3  ft.  4'.  Ans.  15  ft.  8'  0"  8'". 

■i    3.  How  many  feet  of  boards  will  be  required  to  construct  50 

boxes  2  ft.  3'  long,  2  ft.  3'  wide,  1  ft.  11'  high,  making  no  allowance 

for  thickness  of  boards  ?  Ans.  1368  ft.  9'. 

4.  Which  will  contain  more,  and  how  much  more,  a  box  3  ft. 
9'  by  1  ft.  6'  by  2  ft.  3',  or  a  box  2  ft.  6'  each  way  ? 

Ans.  The  latter  by  2  ft.  11'  7"  6"'. 

5.  What  will  be  the  cost  of  polishing  a  piece  of  marble  on  one 
side  and  all  the  edges  at  $.33^  per  square  foot,  the  marble  being 
3  ft.  T  by  1  ft.  9'  and  1'  thick?  .    Ans.  $2.38/^^- 

6.  How  many  cubic  feet  of  masonry  in  a  wall  16  rods  long, 
8  ft.  9'  high,  and  2  ft.  2'  thick  ?  Ans.  5005  ft. 


148  COMPOUND  DENOMINATE  NUMBERS. 

7.  What  is  the  cost  of  laying  two  floors,  each  16  ft.  8'  by  12  ft. 
6',  at  18  cts.  per  sq.  yd.  ?  Ans.  $8.33^. 

8.  Find  the  price,  at  $24  per  thousand  ft.,  of  3  boards  measur- 
ing as  follows:  17  ft.  11'  by  1  ft.  2',  19  ft.  4'  by  1  ft.  11',  and  22 
ft.  8'  by  1  ft.  9'.  Ans.  $2,343. 

9.  How  many  feet,  board  measure,  in  6  planks  2  in.  thick,  each 
25  ft.  9'  long,  6'  wide?  (See  Art.  213,  Ex.  43,  note.) 

Ans.  154 J  ft. 
"^2^0,     Division  of  Duodecimals. 

III.  Ex.  A  plat  of  ground  contains  65  ft.  0'  7" ;  its  width  is 
6  ft.  4' ;  w^hat  is  its  length  ? 

Operation.  6  ft.  in  65  ft.  =  10,  10  X 

6  ft.  4'  )  65  ft.  0'  7"(10  ft.  3'  3",  Ans.     6   ft.  4'  =  63  ft.  4',  which, 

63      4'  subtracted  from  the  dividend, 

1      8   7"  =  20'  7"  gives  a  remainder  of  1  ft.  8' 

19'  0"  7"  =  20'  7"  ;  6  ft.  in  20"=  3' 

V  7"=  19"  3'  X  6  ft.  4'  =  19'  0",  which 

19"  0"'    subtracted  from  20'  7"  leaves 

0     0       1'7";  l'7"  =  19";6ft.inl9" 

=  3"  ;  3"  X  6  ft.  4'  =  19".     Hence  the 

Rule  for  Division  of  Duodecimals.  Divide  the  highest 
term  in  the  dividend  hy  the  highest  term  in  the  divisor  ;  the  quotient 
will  he  the  first  term  in  the  answer.  Midtiply  the  entire  divisor  hy 
that  term,  and  subtract  the  product  from  the  dividend.  Divide  as 
before,  and  thus  proceed  till  all  the  terms  of  the  dividend  are  di' 
vided.  Should  there  he  a  remainder,  it  may  be  reduced  to  num- 
bers of  lower  denominations  and  divided,  or  annexed  to  the 
quotient  in  a  fractional  form,  having  for  its  denominator  the 
divisor  expressed  in  units. 

Examples. 

1.  Divide  54  ft.  7'  4"  6'"  by  4  ft.  1'.  Ans.  13  ft.  4'  6". 

2.  What  is  the  width  of  a  table,  4  feet  3'  long,  which  contains 
14  ft.  2'?  ^ns.  3  ft.  4'. 

3.  How  many  feet  of  joist,  4  inches  wide  and  3  inches  thick, 
allowing  nothing  for  waste  by  sawing,  can  be  made  from  a  piece 
of  timber  44  ft.  5'  long,  1  ft.  3'  wide,  and  1  ft.  4'  thick  ? 

Ans,  888  ft.  41 


REVIEW.  149 

4.  How  many  blocks  of  stone  containing  1  ft.  11'  6"  6'"  can 
be  sawed  from  a  block  containing  11  ft.  8'  9"?         Ans.  6  blocks. 

5.  "What  is  the  thickness  of  a  block  of  granite,  one  of  whose 
surfaces  contains  75  ft.  10'  8",  and  whose  solid  contents  are  107 
ft.  6'  1"  4'"  ?  Ans,  1  ft.  5'. 

— j-SSl,    General  Review,  No.  4. 

1.  Reduce  7  £  3  s.  6  d.  to  farthings. 

2.  Reduce  4876  gr.  to  lb.,  etc.,  Troy. 

3.  4tb,i)S,75,29,8gr.  +  3ib,6§,23,29,8gr.rr:? 

4.  3T.  Icwt.  2qr.  1  lb.  — 1  T.  2  cwt.  3  qr.  7  lb.  8dr.  =  ? 

5.  Im.— 6f.  16r.  3yd.  1  ft.  8in.  =  ? 

6.  Multiply  2  m.  30  ch.  12  1.  by  8. 

7.  Multiply  5  y.  212  d.  10  h.  15  m.  by  20.  (365^  days  to  the 
year.) 

8.  Divide  4  A.  3  R.  24  r.  by  9. 

9.  In  |-  c.  1.  how  many  feet  ? 

10.  What  part  of  1  A.  is  3  R.  13  r.  5  J  ft.  ? 

11.  Reduce  f  cu.  yds.  to  feet  and  inches. 

12.  Reduce  8'  53^"  to  the  fraction  of  a  degree. 

13.  What  cost  12  bu.  2  pks.  of  plums  at  $.06  a  pint? 

14.  What  cost  2  qts.  1^  pts.  oil  at  $1.12  per  gallon? 

15.  Required  the  number  of  square  feet  in  a  garden  4  rds 
long  and  1  rd.  15  ft.  wide. 

16.  How  many  cu.  ft.  of  space  in  a  cellar  measuring  on  th^ 
inside  of  the  wall  5  yd.  1  ft.  in  length,  4  yds.  in  width,  and  10  ft 
in  depth  ? 

17.  What  is  the  difference  of  time  in  two  places  whose  longi 
tudes  differ  7°  8'  4"? 

18.  When  the  difference  of  time  is  3  h.  4  m.  6  s.,  what  is  thto 
difference  of  longitude  between  two  places  ? 

19.  How  many  days  from  Jan.  5,  1864,  to  March  3,  1865  ? 
1^*  For  changes,  see  Key, 


150  DECIMAL  FRACTIONS. 


DECIMAL    FRACTIONS. 


3S9,  As  by  the  Decimal  System  of  representing  numbers 
(Art.  23),  each  lower  denomination  is  one  tenth  of  the  next  higher, 
one  ten  being  one  tenth  of  one  hundred,  one  unit  one  tenth  of  one 
ten,  so  one  unit  may  be  divided  into  ten  equal  parts,  or  tenths, 
one  tenth  into  ten  equal  parts,  or  hundredths,  etc.  Thus  we  have 
fractional  numbers  descending  from  the  unit  by  a  scale  of  tens. 
Represented  as  common  fractions,  the  denominators  of  these 
numbers  are  10,  100  (102),  iqoO  (10^),  etc.     Hence, 

S^3.  A  Decimal  Fraction  is  a  fraction  whose  denominator 
is  some  integral  power  of  ten. 

224L,  Decimal  Fractions  are  generally  written  like  whole 
numbers ;  they  are  distinguished  from  whole  numbers  by  having 
the  decimal  point  placed  at  their  left. 

225»    Decimal  Fractions  are  read  like  whole  numbers,  the 
denomination  being  always  given  ;  this  is  determined  by  the  plact, 
of  the  right  hand  figure  in  reference  to  the  decimal  point ;  thus, 
.5         is  read  5  tenths. 
.05       "     "      5  hundredths. 
.748     «     «     748  thousandths. 
.0748  "     "      748  ten-thousandths. 
7.48       "     "      7  and  48  hundredths. 

SS6.   Numeration  Table. 


^^^.£3^^-M  -^^rrt-S-fS^^^^ 


r£5 


4976542.830471592876 

INTEGERS.  FRACTIONAL  NUMBERS. 


NOTATION  AND   NUMERATION.  151 

Exercises  upon  the  Table. 

1.  Which  place  at  the  right  of  the  decimal  point  is  occupied  by 
tenths  ?  by  thousandths  ?  by  millionths  ?  by  billionths  ?  by  trillionths  ? 
by  hundredths  ?  by  ten-thousandths  ?  by  hundred-thousandths  ?  by 
ten-millionths  ?  by  hundred-millionths  ?  by  hundred-biliionths  ? 

2.  What  denomination  occupies  the  second  place  at  the  right  of  the 
point?  the  thu'd?  the  fourth?  the  fifth?  the  sixth?  the  first?  the 
seventh?  the  ninth?  the  eighth?  the  eleventh?  the  twelfth?  the  fif- 
teenth? 

S2'7.    To  read  decimal  fractions,  observe  the  following 
Rule,     Read  the  decimal  fraction  as  if  it  were  a  whole  num- 
ber, giving  it  the  denomination,  of  the  right  hand  figure. 

Exercises. 
Read  or  write  in  words  the  following:  — 

1.  .9.  5.  .095009. 

2.  .469.  6.  .37f 

3.  ,0599,  7.  .0345706. 

4.  .05099.  ^  8.  .00080007. 

Read  the  following,  first  as  mixed  numbers,  then  as  improper 
fractions :  — 


9,  27,5, 

10.  2.75, 

11.  885.47533. 

12.  7000.0005. 


13.  .7005, 

14.  175.87^. 

15.  250.00554. 

16.  2505.00|.  ' 


Note.  —  The  word  units  maybe  placed  after  the  7000  in  Ex.  12,  in 
reading  it  as  a  mixed  number,  to  distinguish  it  from  the  7  thousand  ten- 
th© usaadths  in  Ex,  13.     Head  thus  in  all  similar  cases  of  ambiguity. 

Name  the  terms  in  the  above  examples,  heginning  at  the  left. 

Ans.  (Ex.  l)  ^  tenths;  (Ex.  2)  4  tenths,  C  hundredths,  9 
thousandths ;  etc, 

S38.    To  write  Decimal  Fractions,  observe  the  following 
Rule.      Write  the  figures  as  in  whole  numbers,  putting  the 
deeirmd  point  so  that  the  right  hand  figure  shall  he  in  the  place  of 
the  denomination  named  in  the  decimal  fraction,  supplying  vacant 
places,  if  there  he  any,  with  zeros. 


152  DECIMAL  FRACTIONS. 

Exercises. 
"Write  tlie  following  in  figures  :  — 

1.  Sixty-four  hundredths. 

2.  -Nine  hundred  forty-two  thousandths, 

3.  Nine  hundred  forty-two  ten-thousandths. 

4.  Eight  thousand  three  hundred  twenty-five  ten-thousandths, 

5.  Seventy-five  hundred-thousandths. 

6.  Seven  thousand  five  hundred-thousandths. 

7.  Fifty  and  four  hundred  eighty-two  thousandths. 

8.  One  hundred  fifty-five  millionths. 

9.  One  hundred  units,  and  fifty-five  miUionths. 

10.  Three  hundred  thousand  eight  hillionths. 

11.  Three  hundred  tiiousand  units,  and  eight  hillionths. 

12.  Forty  million  eight  hundred  four  thousand  and  twenty- 
five,  and  thi*ee  hundred  four  thousand  eight  hundred  seventy-five 
hundred-million  ths. 

13.  Seven  million  units,  and  one  ten-millionth. 

14.  Seven  million  and  one  ten-millicmths. 

15.  Thirty  and  six  tenths. 

16.  Three  hundred  six  tenths. 

17.  Tliree  hundred  seventy  and  f  ten-thousandths. 

18.  Four  hundred  seven  thousand  eight  hundred  seventy-fiv* 
and  ^  ten-billionths. 

Note.  —  Zeros  may  be  annexed  or  omitted  at  the  right  of  a  decimal 
fraction  without  altering  the  value  of  the  fraction,  for  both  numerator  an^ 
denominator  are  thereby  multiphed  or  divided  by  the  same  number.  (Art. 
119,  Prop,  iii.,  iv.)     Thus,  .50  (^%)  3i:.5  (/^)  zz:  .500  (^^0%). 

Fundamental  Operations. 

3S9.  Decimal  Fractions  may  be  written  and  operated  upon 
like  common  fractions,  the  same  principles  being  applicable  to 
both  j  but  as  they  increase  and  decrease,  like  w^hole  numbers,  by 
a  scale  of  tensy  they  can  also  be  treated  in  all  respects  like  whole 
numbers.  Close  attention  must  be  given  to  placing  the  decimal 
point. 


ADDITION.  153 

^30.     Addition. 
To  add  Decimal  Fractions,  observe  the  following 
Rule.    Place  the  figures  of  the  same  denomination  under  each 
other  ;  then  add  as  in  whole  numbers,  observing  to  place  the  deci- 
mal point  in  the  amount  under  those  in  the  example. 

Examples. 


Ans. 


1. 

2. 

3. 

6782.2 

8.752 

5.3125 

298.98 

975.84 

807.06848 

4400.64 

35.075 

9.0875 

3034.05 

780.136 

975.00625 

14515.87 

4. 

5. 

6. 

1482.9 

.594 

875325.075 

29.7868 

8.594 

8753.25075 

668.47 

3.75 

87.5325075 

4872.001 

.674 

875.325075 

8569.8456 

600.044 

8.75325075 

762.4 

600.00449 

39.07528 

6847.9773 

85.8585 

39075.28 

9320.7685 

30.5 

3907.528 

7.  8.75  +  90.095  +  840.6007  +  4  +  67304.745  +  190075. 
40007  +  4006.87  -f  475.44 1=  what  ? 

8.  4  hundred,  and  847  thousandths  -\-  9  thousand  875  and  4 
thousandths  -f-  3  hundred  7  and  3  hundred  7  ten-thousandths-^  6 
thousand  200  units,  and  62  ten-thousandths  =i  what  ? 

9.  9  hundred  units,  and  9  hundredths  -f-  9  thousand  874  and  9 
thousand  874  ten-thousandths  +  987,  and  49  thousand  874 
hundred-thousandths  -\-  9,  and  8  million  749  thousand  874  ten- 
millionths  -\-  98  thousand  749,  and  874  thousandths  -\-  62  thou- 
sand units  and  62  thousandths  =z'i 

10.  205  thousandths  -\-  1  thousand,  and  1  thousand  5  ten-thou- 
sandths -\-  9  hundred  4  hundred-thousandths  -\-  9  million  407 
thousand  units,  and  327  hundred-thousandths  --[-  3  thousand  27» 
and  4  hundredths  =.  ? 


154  DECIMAL  FRACTIONS. 

33 1  •       SUBTKACTION. 

To  subtract  decimal  fractions,  observe  the  following 
Rule.      IVrite  the  subtrahend  beneath  the  minuend,  units  under 
units,  tenths  under  tenths,  etc.;    subtract  as  in  whole  numbers, 
placing  the  decimal  point  in  the  remainder  wider  those  in  the 
minuend  and  subtrahend. 

Examples. 

1.  2.  3. 

From  756.875  56.8507  6.005 

Take     97.486  38.193  .02983 


Ans.  659.389  Ans.  5.97517 

Note. — In  the  3d  example,  as  there  are  no  hundred-thousandths  to 
subtract  from,  and  no  ten- thousandths,  we  reduce  one  of  the  thousandths 
to  ten- thousandths,  and  one  of  the  ten-thousandths  to  hundred-thou- 
sandths, and  subtract. 

4.  From  132.0064  take  123.887. 

5.  Find  the  difference  between  30.801  and  308.01. 

6.  275.87  —  37.15956  =  ? 

7.  From  2  hundred  units  and  5  thousandths,  take  209  thou^ 
sandths. 

8.  The  subtrahend  being  784,  and  20  thousand  456  hundred 
thousandths,  and  the  minuend  906,  and  34  hundredths,  required 
the  remainder. 

9.  From  two  thousand  take  two  thousandths. 

10.  Subtract  24073  thousandths  from  24,  and  73  thousandths. 

11.  What  is  the  value  of  45  million,  minus  45  millionths  ? 

933.     Addition  and  Subtraction. 

1.  The  difference  between  two  numbers  is  67.97,  the  less  being 
9874.08  ;  what  is  the  greater  ? 

2.  The  difference  between  two  numbers  is  29.875,  the  larger 
being  1909  ;  required  the  smaller. 

3.  The  compound  interest  of  a  certain  sum  being  $1.4416, 
exceeds  the  simple  interest  by  $.91375  ;  what  is  the  simple  inter- 
est ? 


DECIMAL  FRACTIONS.  155 

4.  From  G4.0125-f- .09778  take  640125 —.09778. 
1^  For  Dictation  Exercises,  see  Key. 

333,     Multiplication  and  Division  by  10, 100, 1000,  etc. 

Since  numbers  increase  from  right  to  left  by  a  scale  of  tens, 
and  decrease  from  left  to  right  in  the  same  manner,  it  follows 
that 

Ati^  decimal  number,  whether  a  fraction  or  a  whole  number,  may 
he  multiplied  by  10,  100,  or  any  -power  of  \^,  hy  removing  the  dec- 
imal point  as  many  places  towards  the  right  as  there  are  zeros  in 
the  multiplier. 

Thus,  .3  X  10  =  3. ;  .3  X  100  i=  30. ;  .225  X  100  =  22.5. 

^34,     It  follows,  also,  that 

Any  decimal  number,  whether  a  fraction  or  a  whole  number, 
may  be  divided  by  10,  100,  or  any  power  of  10,  by  removing  the 
decimal  point  as  many  places  towards  the  left  as  there  are  zeros  in 
the  divisor. 

Thus,  7  -MO  —  .7 ;  .7  -^  100  :=  .007 ;  78.4  -^  1000  r=  .0784. 

Illustrative   Examples. 


Multiply  50.7  by  10  ;  4.75 
by  100;  13.57  by  1000;  .375 
by  10  ;  and  give  the  sum  of  the 
products. 

Operatiox,  Opeeation, 

50.7      X       10 1=      507.  58.    -f-      10  r=    5.8 


Divide  58  by  10  ;  4.7  by  100; 
83.2  by  1000  ;  18470  by  1000; 
and  give  the  sum  of  the  quo- 
tients. 


4.75    X     100^      475.  4.7 -^-    100=      .047 

13.57    X  1000  —  13570.  83.2  -^  1000  z=z      .0832 

.375  X       10  =         3.75  18470.    -^  1000  =  18.470 


Sum  of  Products,  14555.75  Sum  of  Quotients,  24.4002 

335.     Examples. 
Add  the  following : 

1.  4.75  X  100;  5.84  X  10;  463  X  10.  Ans,  5163t4. 

2.  .031  X   1000 ;  76.218  X  100 ;  4.0005  X  1000 ;  .000987 
X  100000.  Ans.  11752. 

3.  74.7  -r  10  j  16,75  -i- 10  ;  87  -J-  100 ;  1324  -^  1000. 

Ans,  11.339. 


156  DECIMAL  FKACTIONS^ 

4.  756.7  -J-  1000;  20.09  ~  100;  1800  ~  100;  175.005  -x 
10  ;  397000  -^  10000  ;  .5  -^  1000. 

Note.  —  Retain  the  separate  results  in  each  of  the  following  examples, 
and  find  their  sum. 

5.  Divide  182  by  10  ;  multiply  that  result  hy  100;  divide  that 
by  1000  ;  multiply  that  by  10,  and  that  by  10. 

Opekation. 

182  -^  10  =  18.2  ;  18.2  X  100  =  1820. 

1820  ~  1000  z=  1.82 ;  1.82  X  10  ==  18.2;  18.2  X  10  =  182. 

18.2  -f  1820.  4-  1.82  4-  18.2  +  182  =  2040.22,  Ans. 

6.  Divide  796  by  10 ;  divide  that  result  by  100 ;  multiply  that 
by  10.  Sum,  88.356. 

7.  Divide  8394  by  10;  take  y^-^  of  the  quotient;  1000  times 
that  result;  j^^  of  this  product;  y^y  of  this;  divide  this  by  1000, 
and  take  100  times  this  quotient.  Sum,  9334.975794. 

8.  Multiply  .648  by  100 ;  divide  the  product  by  10 ;  multiply 
the  quotient  by  1000 ;  take  ^  of  jV  of  that  product,  and  }  of 
■j^Q  of  this.  Sum,  6876.36. 

^30.   Multiplication. 
III.  Ex.,  I.     Multiply  1.87  by  .5. 

Opkration.       If  1.87  be  multiplied  by  5,  the  product  will  be  of  the 

1.87      same  denomination  as  the  multiplicand,  or  9.35 ;   but 

•^      since  the  multiplier  is  5  tenths,  a  number  but  one  tenth 

Ans.  .935      as  large  as  5,  the  product  will  be  but  one  tenth  as  large, 

and  the  decimal  point  must  be  put  one  place  farther  to  the  left,  making 

the  answer  .935. 

III.  Ex.,  II. 
Multiply   .012         '012  multiplied  by  13=  156  thousandths  (.156)  ; 
by       .13     .012  multiplied  by  .13,  a  number  one  one-hundredth 
Ans.  .00156     ^^  ^^8^  ^^  1^»  ^^^^^  gi^®  ^  product  one  one-hundredth 

as  large,  or  .00156. 
From  these  illustrations  we  derive  the  following 
Rule.     Multiply  as  in  whole  numbers,  and  point  off  as  many 
places  for  decimal  fractions  in  the  product  as  there  are  places  of 
decimal  fractions  in  both  the  factors.     If  there  are  not  figures 
enough  in  the  product  prefix  zeros. 


DIVISION.  157 

ExAMPLEi. 

1.  .92  X  5.6  =  ?  Ans.  5.152. 

2.  9.72  X  .87  =  ?        ■  Ans,  8.4564. 

3.  .687  X.038  =  ? 

4.  95.874  X  4.007  =:  ? 

5.  308.  X  .0063  :=  ? 

6.  .000001  X  1000000  r=? 

7.  4.02  X  400.02  X  402.01  =  what? 

8.  What  is  the  cost  of  whitewashing  the  ceiling  of  a  room 
18.75  yds.  long,  10.82  yds.  wide,  at  $.015  per  sq.  yd.? 

9.  What  cost  11  thousand  bricks,  at  $12.3175  per  M.? 

10.  What  cost  635  laths,  at  $.276  per  hundred  ? 

11.  What  is  the  cost  of  tiling  a  roof  289  feet  long  and  54  feet 
wide,  at  $8.25  per  hundred  feet. 

12.  What  will  be  the  cost  of  shingling  the  above  roof  with 
shingles  which  cost  $6.50  per  thousand  feet,  the  shingles  lying 
one  third  to  the  weather  ? 

1^^  For  Dictation  Exercises,  see  Key. 

S37.   Division. 
III.  Ex.,  I.     Divide  3.864  by  12. 

Operation.        Since,  in  the  above  example,  we  divide  3684  thou- 
12  )  3.864     sandths  into  12  equal  parts,  it  is  evident  that  the  quotient 
Ans.  .322     will  be  thousandths,  and  require  three  decimal  places. 
Therefore,  when  the  divisor  is  a  whole  iiumber,  the  quo- 
tient must  be  of  the  same  denomination  as  the  dividend. 

III.  Ex.,  II.     Divide  1.224  by  .36. 

Operation.  Here  the  divisor  is  not  a  whole  number, 

i36.)1^22.4(3.4,  Ans.     but  hundredths.     It  may  be  made  a  whole 

1  OR  •  . 

^  number  by  removing  the  decimal  point  two 

I  J^^  places  to  the  right.     If  we  also  remove  the 

—  decimal  point  in  the  dividend  two  places  to 

the  right,  the  divisor  and  dividend  will  be 

equally  multiplied,  and  the  quotient  resulting  from  the  division  will  be 

the  same  as  if  no  alteration  had  been  made  (Art.  119,  Prop.  111).     The 

dividend  now  being  divided  by  a  whole  number,  the  quotient  must  be 

of  the  eame  denomination  as  the  altered  dividend,  or  tenths. 


158 


DECIMAL  FRACTIONS. 


From  the  above  illustrations  we  derive  the  following 

Rule.  To  divide  decimal  fractions :  Divide  as  in  whole 
numbers.  If  the  divisor  is  a  whole  number,  point  off  as  many 
decimal  places  in  the  quotient  as  there  are  decimal  places  in  the 
dividend.  If  the  divisor  is  not  a  whole  number,  make  it  a  whole 
number  before  dividing,  by  removing  the  decimal  point  to  the  right 
Remove  the  decimal  poiiit  in  the  dividend  as  many  places  to  tht 
right;  divide,  and  point  off  as  many  decimal  places  in  the  quo- 
tient as  there  are  in  the  altered  dividend. 

Note  I.  —  When  there  is  a  remamder  after  all  the  figures  in  the  divi- 
dend are  exhausted,  zeros  may  be  annexed,  and  the  division  continued.  In 
pointing  off,  the  annexed  zeros  must  be  considered  as  places  in  the  div- 
idend. 

Note  IE.  —  In  the  examples  in  this  book,  when  there  is  a  remainder,  the 
quotient  mav  be  continued  to  the  fifth  decimal  place,  if  no  other  direction 
is  given. 

Examples. 


1.  14.91 -^  7  =  ?       J[^.  2.13. 

2.  .072  -^  6  =?         Ans..0l2. 

3.  8.25  -^  1.5  =z  ?         Ans.  5.5. 

4.  3.24  -f.  .81  =  ? 

5.  .00468  4- .013  =? 

6.  5446.776  -^  8  =  ? 

7.  180.375  ~  1.625  =  ? 

8.  579  ~  .075  =  ?    Ans.  7720. 

9.  6.9705  ~  .45  =  ? 

10.  .0033  4- .011  =  ? 

11.  1.29  -^.32=? 

Ans.  4.03125. 

12.  .705  4-7.5=:?    Ans.  .094. 

13.  3-^29.9=?^«s..l0033+. 

14.  20  4- .013  =  ? 

Ans.  1538.46153+. 

15.  4066.2  -f-  .648  =  ? 


16.  68077^71.66=?^7i5.950. 

17.  .880351  -^  897  =  ? 

18.  .1706  ^4.2368  =  ? 

19.  56.28  -T-  .0056=? 

20.  10588.1  -^  .4606  =  ? 

21.  .417196  ~  58.76  =  ? 

22.  .08  -^  1.611  =? 

23.  24000-^  1.1713=? 

24.  1.3  -^-  197.59  =  ? 

25.  828.45  -f-  26.3719=? 

26.  25.25  -f- 42993.78=? 

27.  1203.488  -^  28.6=? 

28.  49.2654756  -i-  .0759  =? 

29.  2464.176 -f- 57.2=? 

30.  164.6156-^  1334  =  ? 

31.  .07991997  ^  83497=? 

32.  20339.82009  -f- 1.07001  =? 


33r  Divide  93.75  by  3265096.575,  and  give  three  significant 
figures  in  the  quotient. 


REDUCTION.  159 

34.  Find  the  product  of  the  quotients  of  the  following  to  6 
places :  .65084958  -^  3.69  ;  40  -^  5000. 

35!  What  is  the  quotient  of  1.497  -^  (260.401  —  13.02)  ? 

36.  Required  the  product  of  the  quotients  of  the  following: 
1021  ten  millionths  -^  107  ten  thousandths;  2012  millionths  -^ 
1.006. 

37!  Divide  600  by  .006,  multiply  the  quotient  by  .05,  and  by 
that  product  divide  .005. 

38!  (1  -^  .002)  X  (.2  ^  50)  r=  ? 

39!  (80.481825  -^  89.325)  X  (9617.5168  ^  .47896)  r=? 

40!  Required  the  product  of  the  sum  and  difference  of  the 
following :  856494  -^  839.7 ;  .0094658  H-  9.4.     (To  6  places.) 

4l!   Divide  the  difference  of  the  above  quotients  by  their  sum. 

^^  For  Dictation  Exercises,  see  Key. 

338.     Reduction   of   Common   Fractions   to  Decimal 
Fractions. 
III.  Ex.     Reduce  |  to  a  decimal  fraction. 

Operation.  |=J  of  7  ;  ^  of  7:=  no  whole  ones,  with  a  re- 

8  )  7.000  mainder  of  7,  which  reduced  =  70  tenths  (7.0) ; 

\  of  70  tenths  :=.  .8  with  a  remainder  of  .6  ;  .6  zz: 

'         *         60  hundredths;  |  of  .60 =.07  with  a  remainder  of 

.04  ;  .04=40  thousandths ;  i  of  .040  =  .005  ;  .'.  |  =  .875.    Hence  the 

Rule.     To  reduce  a  common  fi-action  to  a  decimal  fraction: 

Annex  zeros  to  the  numerator,  and  divide  it  by  the  denominator. 

Point  off  as  inany  decimal  places  as  there  are  zeros  annexed. 

Examples. 

Reduce  to  decimals, 


1.  |.         Ans.  .375. 

2.  /g.        Ans.  .35. 

3.  yf  ^.    Ans.  .024. 

5.  f  |.  Ans.  3.0625. 

6.  5|.    Ans.  5.125. 


7.  W- 

8.  1-e^. 

9.  83.      Ans.  8.75. 

10.  17yVV 

11.  i.oo^v 

Ans.  1.0012. 

16.  Reduce  to  decimals,  and  add,  I,  ifffi,  3^  J^. 

17.  Reduce  to  seven  places,  and  add,  1.82^^,  .009y^^,  aad 

^*  For  Dictation  Exercises,  see  Key. 


12.  .0-^V 

Ans.  .00125. 

13.  I.    Ans.  .166-f-. 

14.  ^V 

15.  tV 


160  DECIMAL  FRACTIONS. 

339,    Reduction   of   Decimal   Fractions   to    Common 

Fractions. 

III.  Ex.     Reduce  the  following  to  common  fractions:  .75; 

.0125  and  6.25 

Operatiox. 

•'^5z=:    ^\^^    —  f,   Ans. 

6.25  —  e^%  =  61,  Ans. 
Hence  the 

Rule.  To  reduce  a  decimal  fraction  to  a  common  fraction : 
Represent  the  decimal  fraction  in  the  form  of  a  common  fraction 
having  for  its  denominator  1  with  as  many  zeros  annexed  as  there 
are  decimal  places  in  the  decimal  fraction,  and  reduce  the  common 
fraction  to  its  lowest  terms. 

Examples. 

Reduce  to  common  fractions, 


1.  .0625.    Ans. 

2.  .0025. 
.3.  .00064. 


4.  4.0875.  Jws.W^. 


7.  3.1f 

8.  1.0-^. 

9.  1.0068933if 


5.  .08J.       Ans.-j^^' 

6.  .15^^^. 

^^  For  Dictation  Exercises,  see  Key. 

340 •  To  add  or  subtract  decimal  fractions  terminated  by 
common  fractions  :  Reduce  all  the  decimals  to  the  same  denomina- 
tion ;  then  add  or  subtract  as  by  Art.  143  ayid  144 ;  thus,  .3^^  + 
.83^  =  what  ?     .3^  +  .831  —  ,^^  4.  .83§  —  1.1 6|,  Ans. 

Examples. 

1.  Add  .087^,  9.0^,  .7^,  275|,  and  .Oj\.        Ans.  285.2549/^.' 

2.  Add  19.37i,  lO.O^^,  and  .041 6|. 

3.  Subtract  .05555f  from  .3333^. 
4!  Subtract  1.207624f  from  jf. 

5.*  .31  +  .6§  +  .831  _[_  .285714f  +  .571428f  +  .63^^  =  ? 

341*,    Circulating  Decimals. 

If  the  denominator  of  a  common  fraction  (when  the  fraction  is  in  its 
lowest  terms)  contains  any  prime  factor  besides  2  and  5,  the  fraction  is 
not  capable  of  being  entirely  reduced  to  a  decimal  form. 


CIRCULATING  DECIMALS.  161 

In  reducing  such  fractions,  if  the  division  be  c  ntinued,  the  same  fig- 
ures will  recur  again  and  again  in  the  decimal  fraction.  These  fractions 
are  called  Repeating  or  Circulating  Decimals.  The  figures 
which  repeat  are  called  a  Repetend, 

A  Repetend  is  distinguished  by  two  dots  written  over  the  first  and 
last  of  the  figures  that  repeat ;  thus,  \\  z=z  .297297-f-  =  .297. 

34^t    Examples. 
Reduce  to  decimal  fractions, 


1.  i. 

Am.  .3  J  or  .3. 

4-  h  h  \h 

2.  |. 

Am.  .6|  or  .6. 

5-  i  A.  aV- 

3.  |. 

Ans.  .83J  or  .83. 

6.  t\,  ^"1,  A- 

34:3t  Reduction  of  Circulating  Decimals  to  Com- 
mon  Fractions. 

It  can  be  proved  that  the  Repetend  of  a  Circulating  Decimal 
equals  a  fraction  whose  numerator  is  the  repetend,  and  whose 
denominator  is  as  many  9's  as  there  are  places  in  the  repetend. 
Hence  the 

Rule.  To  reduce  a  Circulating  Decimal  to  a  common  fraction : 
JExpress  the  repetend  as  a  common  fraction  having  as  many  9*5 
for  the  denominator  as  there  are  figures  in  the  repetend,  and  re- 
duce. If  any  part  of  the  decimal  fraction  does  not  repeat,  annex 
the  reduced  repetend  to  it,  and  change  the  complex  fraction  thus 
obtained  to  a  simple  fraction. 

Note.  —  Circulating  decimals  may  be  added,  subtracted,  multiplied, 
and  divided,  by  first  reducing  them  to  common  fractions.  Other  pr«cessei! 
might  here  be  given,  but  the  reasoning  is  too  abstruse  for  an  elementary 
treatise. 

III.  Ex.,  I.     Reduce  .09  to  a  common  fraction. 

9  ,       . 

Operation.  .09  ^-^'qq  —  Tt>  -4^*' 

III.  Ex.,  II.      Reduce  .16  to  a  common  fraction. 

1#       If 
Operatiox.    .16z=:-^ziz— z=J^.  Ans. 
10       10       6' 

Examples. 
Reduce  the  following  to  common  fractions : 
11 


L  .6. 

Ans.  |. 

2.  .83. 

Ans.  I. 

a  .1881 

162  DECIMAL  FRACTIONS. 

4.  .428571. 

5.  .714285. 

6.  .2142857. 

944,     To   Reduce  Compound   Numbers   to  Decimal 

Fractions  op  Higher  Denominations. 

III.  Ex.,  I.    Reduce  2  d.  3  qr.  to  the  decimal  of  a  shilling. 

Operatiox.  Since  4  qr.  equal  1  d.,  there  will  be  \  as  many  d. 

4  3.00  qr.  as  qr.,  or  |  d.,  which  equals  .75  d. ;  this,  with  the 

12  2  75000  d  ^  ^*  S^^^^^»  equals  2.75  d. ;  since  12  d.  equals  1 

-. ■     shilling,  there  will  be  ^  as  many  shillings  as  d., 

III.  Ex.,  II.  What  is  the  value  of  3  rds.  4  yds.  2  ft.  in  the 
decimal  of  a  rod  ? 

Operation.  gi^ce  3  ft.  equal  1  yd.,  there  will  be  ^  as 

2.00000  It.  many  yds.  as  feet,  or  |  yds.,  which  equals 

4.66666  4- yd.  ^g  yds.,  this,  with  the  4  yds.  given,  equals 

4.6  yds. ;  since  54  yds.  equals  1  rod,  there 

9.33333+ half  yd.  ,         i  !?  ^  i 

3^4848  + rods,  J[n..     ^'^^^  ^^  5^  °"  ^'^  ^'  ^^"^  ^'^^^  ^'  ^^^"  ^"- 

From  the  above,  we  deduce  the  following 

Rule.  To  reduce  compound  numbers  to  decimal  fractions  of 
higher  denominations :  Divide  the  number  of  the  lowest  denom-^ 
ination  by  what  it  takes  of  ifiat  denomiiiation  to  make  one  of  the 
next  higher  ;  'place  the  quotient  as  a  decimal  fraction  at  the  right 
of  that  higher ;  so  continue  till  all  the  terms  are  reduced  to  the 
denomination  required. 

Examples. 

1.  Reduce  7  d.  3  qr,  to  the  decimal  of  a  £. 

A)is.  £.03229+. 

2.  Reduce  3  da.  22  h,  4  m.  48  sec,  to  the  decimal  of  a  week. 

Ans,  .56  wk. 

3.  Reduce  5  cwt.  3  qr,  10  lb,  to  the  decimal  of  a  ton. 

4.  Reduce  5  cord  ft.  12  cu.  feet  to  the  decimal  of  a  cord, 

5.  Reduce  10  oz.  5  pwt.  12  gr.  to  the  decimal  of  a  pound. 

6.  Reduce  80  cu,  ft.  to  the  decimal  of  a  cord. 


REDUC110JS  163 

7.  What  is  the  value  of  2  fur.  7  rd.  10  ft.  expressed  in  th« 
decimal  of  a  mile  ? 

8.  What  part  of  a  ream  is  15  quires  12  sheets  ? 

9.  What  part  of  an  acre  is  3  R.  15  rd.  6  yd.  83  ft.  ? 

10.  Reduce  7  S.  8°  5'  38"  to  the  decimal  of  a  great  circle. 
I^^  For  Dictation  Exercises,  see  Key. 

345.  To  Reduce  Decimal  Fractions  of  Higher  De- 
nominations TO  Whole  Numbers  op  Lower  Denomina- 
tions. 

III.  Ex.     Reduce  .13125  lbs.  Troy  to  oz.,  &c. 
Opkration. 
lb.  .13125  Since  12  oz.  =  1  lb.,  there  will  be  12  times  as 

]^  many  ounces  as  pounds,  =z  1.575  oz. ;    since  20  pwt. 

oz.  1.575  m  1  oz.,  there  will  be  20  times  as  many  pwt.  as 

^  ounces,  =z  11.5  pwt. ;  since  24  gr.  :=.  1  pwt.,  there 

pwt.  11.5  will  be  24  times  as  many  grains  as  pwt.,  rz:  12  gr. 

^"^  Ans.  1  oz.,  11  pwt.,  12  gr.    Hence  the 
gr.  12. 

Rule.  To  reduce  decimal  fractions  of  higher  denominations 
to  whole  numbers  of  lower  denominations :  Multiply  the  decimal 
fraction  hy  what  it  takes  of  the  next  lower  denomination  to  make 
a  unit  of  the  denomination  of  the  given  decimal,  pointing  off  as 
in  multiplicatiori  of  decimals;  so  continue  till  the  number  is 
reduced  as  low  as  is  required. 

Examples. 
Reduce  to  whole  numbers  of  lower  denominations, 

6.  1.0004^  of  a  bushel. 


1.  .8975  of  a  week. 

Ans.  6  d.  6  h.  46  m.  48  s 

2.  5.624  £. 

3.  .0074623  tb. 

4.  .7587565  hhd. 

5.  .375  of  a  fathom. 
^^  For  Dictation  Exercises,  see  Key. 


7.  .319iofabbL  (31  gall.) 

8.  .578  cord. 

9.  .0756  of  a  degree. 

10.  2.834  of  1  solid  yard. 

11.  .086  of  a  Julian  year. 


164  DECIMAL  FRACTIONS. 

S46.     Questions  for  Review. 

What  are  Decimal  Fractions?  How  are  they  generally  written? 
how  read  ?  How  distinguished  from  whole  numbers  ?  Which  figure 
indicates  the  denomination?  What  is  the  name  of  the  first  place 
at  the  right  of  the  point?  of  the  second?  third?  fourth?  fifth? 
sixth  ? 

Which  is  the  place  of  thousandths?  of  millionths?  of  billionths? 
of  trillionths  ? 

Give  the  rule  for  reading  a  decimal  fraction. 

Read  7.05  as  a  mixed  number  j  as  an  improper  fraction. 

Read  .20  and  .21  so  that  they  may  be  distinguished.  Read  .504  and 
500.004. 

Is  the  value  of  a  decimal  fraction  altered  by  annexing  ciphers? 
What  is  changed  ?  Why  does  the  value  remain  the  same  ?  What  is 
the  effect  of  placing  a  cipher  between  the  decimal  fraction  and  the 
point  ? 

Give  the  rule  for  writing  decimal  fractions.  Rule  for  Addition  ;  for 
Subtraction ;  for  multiplying  by  10,  100,  1000,  &c. ;  for  dividing  by  10, 
100,  &c. ;  general  rule  for  multiplication. 

Illustrate  the  rule  by  an  example,  and  give  the  reason  for  pointing 
off. 

Give  the  rule  for  division  of  decimals.  Perform  an  example  to  illus- 
trate the  rule,  and  explain.  When  the  dividend  does  not  contain  the 
divisor  what  must  be  done  ? 

Rule  for  reducing  common  fractions  to  decimals.  Illustrate  and 
explain. 

Rule  for  reducing  a  decimal  to  a  common  fraction.  Illustrate  and 
explain. 

What  fractions  cannot  be  reduced  wholly  to  the  decimal  form  ?  What 
are  they  called  ? 

What  are  the  repeating  figures  called?  How  is  a.repetend  distin- 
guished ? 

Rule  for  reducing  circulating  decimals    to    common  fractions. 

Rule  for  reducing  a  compound  number  to  decimals  of  higher  denom- 
inations.    Illustrate. 

Rule  for  reducing  decimals  to  whole  numbers  of  lower  denomina- 
tions.   Elustrate. 


miscellaneous  examples.  165 

347*     Miscellaneous  Examples. 

1.  What  is  the  amount  of  3.75  tons,  .085  tons,  1.17|  ton^  and 
1  cwt.  3  qr.  ? 

2.  What  will  be  the  interest  on  $585,  for  6  days,  if  the  interest 
)n  $1  be  $.001  ? 

3.  What  is  the  amount  of  $75823  for  7  y.  3  m.  15  d.,  if  the 
amount  of  $1  for  the  same  time  be  $1.510416§? 

4.  At  $1.33J  a  pair,  how  many  cases  of  shoes,  of  63  pairs 
each,  can  be  bought  for  $936? 

5.  How  many  acres  of  land  in  a  lot  which  is  105  rd.  4  yd.  1-^ 
ft.  long,  and  100.356  rd.  wide  ? 

6.  Required  the  price  of  three  boards  at  $.03^  per  sq.  ft.,  the 
boards  being  of  the  following  dimensions :  17.75  ft.  by  1  ft.  3  in. ; 
15  ft.  10  in.  by  1.37^  ft.,  and  13.5  ft.  by  .916§  ft. 

7.  What  is  the  amount  due  for  the  following  ? 

5200     ft.  of  boards  at  $20  per  M. 
700^  ft.  "        «       «     22.50  per  M. 
94    ft.  "        "       «     36  « 

8.  If  3  hhd.  42  gall.  2f  qt.  of  molasses  cost  $92.64,  what  is 
the  price  per  hhd.  ? 

9.  What  is  the  cost  of  board  for  7  y.  10  m.  18  d.,  at  $200  per 
jrear? 

What  cost 

10.  9  gall.  3.4  qt.  of  vinegar,  at  $.12^  per  gall.  ? 

11.  30  ch.  1  rd.  15  1.  of  a  canal,  at  $3550  per  mile? 

12.  5  bu.  2  pk.  3  qt.  of  wheat,  at  $1.25  per  bu.? 

13.  47  gross,  10  doz.  pens,  at  4  s.  6d.  per  gross? 

14.  6  lb.  7  oz.  6  pwt.  7  gr.  of  gold,  at  $16.30  per  oz.  ? 

15.  17f  yds.  of  ribbon,  at  $.19  per  yd.? 

16.  12J-  doz.  chairs,  at  $1.90  apiece? 

17.  A  road  9  m.  3  fur.  12^  rd.  long,  at  $2475  per  mile? 

18.  12520  oranges,  at  $2f  per  hundred? 

19.  3  cwt.  40  lb.  herring,  at  12  s.  6d.  per  cwt.? 

20.  At  $4  per  bu.  how  many  bu.,  pk.  and  qt.  can  be  bought 
for  $l5.37i  ? 


166  PRACTICE. 

21.  At  1  s.  9  d.  per  lb.  what  cost  3580.5  lb.  hides  ? 

22.  If  5  lb.  5  oz.  of  beef  cost  $.564||,  what  is  the  price  per 
lb.? 

23.  Required  the  cost  of  19  gall.  3  qt.  1  pt.  of  oil,  at  2  s.  6  d. 
per  gall. 

24  If  25375  feet  of  boards  cost  $240,555,  what  is  the  price 
per  M.  ? 

25.  What  is  the  cost  of  7  §,  5  3,  2  B,  of  medicine,  at  $.96  per  lb.? 

26.  How  many  cords  in  a  load  of  wood,  6.5  ft.  long,  4.8  ft.  wide, 
and  3.2^  ft.  high  ? 

27.  How  many  casks  gauging  10.485  gall,  can  be  filled  from  a 
hogshead  gauging  83.88  gall.  ? 

28!  What  will  be  the  cost  per  sq.  yd.  if  $157,675  are  paid  for 
laying  4  pieces  of  sidewalk,  measuring  as  follows  :  40§  ft.  by  4 
ft.,  75  ft.  by  7.84  ft.,  8  ft.  10  in.  by  4.5  ft.,  and  100  ft.  by 
18.37 J  ft.? 


PRACTICE. 


248.  Practice  is  the  process  of  finding  the  value  of  a 
quantity  by  operating  upon  an  assumed  value,  or  by  combining 
the  values  of  convenient  parts. 

III.  Ex.,  I.     What  cost  5750  lbs.  tea,  at  37^  cts.  per  pound? 
Operation. 
5750  lbs.  at  $1  per  lb.  will  cost  $5750. 

«       «    "   $.25    "         «     «  ^  of  5750       =     1437.50 
«      «   "   .12^    «         «     "  I  of  1437.50  z=z       718.75 

$2156.25,  Ans. 

OR, 

5000  lbs.  at  .37^    will  cost  5000  X  .375  =:  $1875. 

500    «     «      "         "      "  ^Ig- of  1875      =:        187.50 

250    "     "      «        "      "  4  of  187.50    —         93.75 

5750  lbs.  at  .37^  per  lb.,  =   $2156.25,  Ans> 


PRACTICE. 


16? 


III.  Ex.,  II.     What  is  the  price  of  17  A.  3  R.  25  rds.  of  land, 

at  $200  per  acre  ? 

Operation. 

If  the  price  of  1  acre  is  $200. 


the  price  of  17  acres  will  be  17  X  200  = 

3400. 

«       «      "     2  roods   "     "     ^  of  200  z=i 

100. 

«       «      «     1       «       "     "     1  of  100  =z 

50. 

«       "      "  20  rods      «     "     ^  of    50 1= 

25. 

"       «      «     5     "        "     "  '  ^  of    25  z^z 

6.25 

«      «     «  17  A.  3  R.  25  rds.              — 

$3581.25,  Anjt. 

III.  Ex.,  III.     What  cost  5  T.  13  cwt.  1 

qr.  10  lbs.  of  hay,  at 

$16.67  per  ton? 

Opkration. 

If  the  price  of  1  ton                                        : 

=  $16.67 

the  price  of    5  tons  will  be      5  X  16.67      . 

=    83.35 

"       «      "  10  cwt.     "     "      ^  of  16.67      : 

=1      8.335 

«       "     «     3     "       "     "     ^\  of  8.335      : 

=       2.500+ 

«       "      "    1  qr.        "     "  i  of  ^ig  of  8.335 

=        .208+ 

«       "      "  10  lbs.       "     "        1  of  .208       : 

=:        .083+ 

«        «       «     «  T    IS  rwf.  1   nr-  10  lbs. 

--  <JjQ4.47fi4-.  Jnjf. 

III.  Ex.,  IV.     What  cost  8.96  bbls.  of  flour,  at  $9.87^  per 

l»Hrrel? 

Operation. 

The  cost  of  8.96  bbls.  of  flour,  at  $1        per  bbl.  =:     $8.96 


<( 

u 

« 

it 

ti 

a 

« 

(( 

it 

ti 

ti 

it 

t( 

it 

it 

it 

ti 

it 

$.121 


"     "   =     89.60 
"     "   =       1.12 


11^    "     "   =  $88.48,  Ans. 


III.  Ex.,  V.     What  cost  17  yds.  of  velvet,  at  3£  os.  lOd.  per 

yard? 

Operation. 

£     B.  d. 

17  yds.,  at  3£,  will  cost  17  X  3£  =  51     0  0 

«      "      "  5s.,    "      "     17  X  i£  =   4     5  0 

"      "      "   lOd., "      "     17  X  |s.—    0  14  2 

17  yds.,  at  3£  5s.  lOd.,  will  cost      £55  10  2  Ans, 


(68  PRACTICE. 

349*     Table    of  Aliquot   Parts. 


Of  a  $. 


50 

83i 

25 

20 

16| 

10 

H 

5 
2 


Of  a  £. 

Of  J 

I  Shilling. 

Of  a  Ton. 

8.     d.         £. 

d. 

£. 

Cwt.  qr.       ton. 

10     =  i 

6 

— 

ilio    =  J! 

6  8=;    J 

4 

r= 

4 

5      =    i 

5      =    } 

3 

: — 

i 

4      =    i 

4      ==    i 

2 

— 

i 

2   2=    i 

34=    i 

li 

= 

i 

2      =^ 

2  6=    i 

1 

rzz 

■,\ 

1    l=l\ 

2      =A 

1      =i. 

1      =A 

Of  a 

Cwt. 

Qr.  lb. 

cwt. 

2 

==  i 

1 

=  i 

20 

=  i 

m 

=  i 

10 

=  tV 

6i 

=A 

5 

=ijV 

Of  an  Acre. 


R.    rd.  A. 


=  i 

32=  I 
20=  I 
16  =  ^ 


Note.  —  This  table  can  be  profitably  extended  by  the  pupil  to  other 
denominations,  as  Time,  Length,  etc. 

3^0.     Examples. 
What  is  the  cost  of 

1.  3  T.  13  cwt.  of  hay,  at  $12|  per  ton  .?  Ans.  $45,625. 

2.  13  cwt.  3  qr.  15  lb.  of  cotton,  at  $8.06^  per  cwt.  ? 

Ans.  $112.068f. 

3.  200  yd.  3  qr.  of  sarcenet,  at  $.12^  a  yd.  ?      Ans.  $25.09 1. 

4.  7  lb.  10  oz.  of  tea,  at  $.b^  per  pound  ?      Ans.  $4.289yV 

5.  A  farm  of  40  A.  3  R.  31  rd.,  at  $82.50  per  acre  ? 

Ans.  $3377.859|. 

6.  1872  lbs.  of  butter,  at  16f  cents  per  pound?      Ans.  $312. 

7.  25350  ft.  of  gas,  at  3^  mills  per  foot  ?  Ans.  $84.50. 

8.  5  T.  12  cwt.  2  qr.  6^  lb.  of  hemp,  at  $180  a  ton? 

0.  Fencing  180  rd.  3  yd.  2  ft.  of  road,  both  sides,  at  $1.25 
^rtr  rd.  ? 

10.  Fencing  a  square  lot  of  the  above  length,  at  $.90  per  rd.  ? 

11.  3  pko  7  qt.  of  berries,  at  82  a  bushel  ? 

12.  Insuring  a  house  5  y.  7  m.  20  d.,  at  $6.66f  a  year  ?    (30  d. 
:=  1  m.). 


GENERAL  REVIEW.  169 

13.  12  boxes  of  shoes,  each  containing  48  pairs,  at  $.93J  per 
pair? 

14    17  cd.  7  cd,  ft,  8  cu-  ft,  of  wood,  at  $5.52  per  cord  ? 

15.  Board  for  9  w.  4J-  d.,  at  $4,75  per  week  ? 

16.  7^  doz.  knives,  at  $3,75  a  dozen  ,^ 

17.  3|  yd,  of  silk,  at  $l,12j  per  yard  ? 

18.  Making  7  m,  7  f,  35  rd,  of  raih-oad,  at  $650,000  per  mile.  ? 

19.  G  hhd.  42  gall,  of  wine,  at  $110,15  per  hogshead.? 

20.  75  cwt.  75  lb,  of  tobacco,  at  $18f  per  cwt.  ? 

21.  7  lb.  7  oz,  2  pwt,  IG  gr.  of  silver,  at  $12,87J-  per  pound? 

22.  If  the  interest  of  $1000  for  1  year  is  $105,  what  would  bo 
the  interest  of  tlie  same  sum  for  3  y,  8  m,  24  d,  ? 

23.  What  is  the  amount  of  my  salary  for  3  y,  4  m,  25  d.,  at 
$600  per  annum  ? 

24.  The  rent  of  a  musical  instrument  for  a  certain  time  was 
$60,  the  rent  being  $6  per  quarter ;  what  would  have  been  the 
rent  for  the  same  time  at  $7  per  quarter?  at  $5  ?  at  $8?  at  $3  ? 
at  $9?  at  $10?  at  $7^?  at  $4^?  sit  $12i?  at  $1  per  month? 
at  $1^  per  month  ?  Smn  of  umwers^  $740. 

25.  What  cost  587  lb,  of  soap,  at  1£  8s,  9d,  per  cwt.? 
2Q,  What  cost  17  T.  9  cwt,  3  qr.,  at  5£  3s,  8d,  per  ton? 

Note.  —  For  further  examples  that  mmy  be  performed  by  Practice,  the  < 
pupil  is  referred  to  Art.  247, 

3«>1.    General  Eeview,  No,  5, 

PART    I, 

1.  Multiply  675  by  -j^^,  the  product  hy  100;  divide  the  last 
product  by  iOOO,  this  quotient  by  i,\  multiply  ih.Q  last  quotient 
by  g^gj,  and  add  the  five  results, 

2,  From  ,1  lb.  take  .0678  lb. 
^.  Multiply  8.05^  by  .056|, 

4.  What  is  the  cost  of  « 4  ft,  of  hoards,  at  $20  per  M.  ? 

5.  .007644 -r- 36  =  ? 

6.  8.052-^.0044  =  ? 

7.  10. -r- 1000  =  ? 

8.  .065455  ~  .065  =  ? 

9.  Reduce  ^q  to  a  decimal* 


170  GENERAL  REVIEW. 

10.  .3^+.92|r=? 

11.  What  is  the  cost  of  3  pk.  7  qt.  of  peas,  at  $3.75   per 
bushel  ? 

12.  Reduce  .8765f  deg.  to  whole  numbers  of  lower  denomina- 
tions. 

13.  Reduce  .015  and  .0096  to  cpmmon  fractions- 
14!  Reduce  .39  and  .3432  to  common  fractions. 

PART    II. 

1.  "What  is  the  largest  number  that  will  exactly  divide  475200 
and  216000? 

2.  In  324  sheets  of  lead,  each  12  in.  by  8  in.,  -^  in.  thick, 
how  many  solid  inches  ? 

3.  In  20692  sq.  rd.  how  many  acres  ? 

4.  Reduce  19  h.  12  m.  to  the  fraction  of  a  day. 

5.  What  is  the  5th  power  of  .09  ? 

6.  50.76§  +  .834|  +  .0^r=? 

7.  If  6  yd.  of  cloth  cost  $51 ,  what  will  14f  yd.  cost  ? 

8.  Add^£,  f  s.,  and|d. 

9.  Reduce  .21  pt.  to  the  decimal  of  a  peck. 

10.  How  many  inches  in  length  of  that  which  is  8^  in.  in 
•  breadth  will  make  a  square  foot  ? 

11.  Reduce  yfg-  to  a  decimal  fraction. 

12.  Carry  out  the  following  bill :  — 

Franklin,  Dec.  14,  1864. 
B.  Frank  Watson, 

Bought  of  B.  Cooudge; 

10850  ft.  Boards,  at  $11.       per  M., 

3000  «        «         «     19.375      "  

2500  «  Lathing,  «       4.75        «  

1500  "  Shingles,  '^6.  «  

250  «  Plank,     «     13.  « 

1250  "  Timber,  «     12.80       " 

4220       Bricks,    "     12.50        "  

Received  payment,  ' 

B.    COOLIDGE. 

t^  For  Changes,  see  Key. 


PERCENTAGE. 


171 


PERCENTAGE. 


9«>3.  The  subject  of  Percentage  comprises  operations  in 
hundredths.  Per  cent.,  from  the  Latin  per,  by,  and  centum,  hun- 
dred, signifies  by  the  hundred ;  thus,  4  per  cent,  of  any  number 
c,Y  quantity  is  j^^  of  that  number  or  quantity. 

Any  per  cent,  may  be  expressed  as  a  decimal  fraction,  as  a 
common  fraction,  or  by  the  use  of  the  following  sign,  %  ;  thus, 


4    per 

cent 

is  written  .04      or 

T^^  or  4%. 

H    " 

u 

"         "       .03|     « 

tI^  "   H%' 

lA" 

(C 

"         "       .OlA  "  4t^t.  -    1^V% 

i      " 

ii 

"     "    m  "  j^  -  i%. 

Examples. 

Represent  the  following  rates  decimally : 

— 

1.  8%.       Arts. 

.08. 

4.    13%. 

7.  16|%. 

2.  20%.    Ans. 

.20. 

5.  6i%. 

8.  106%. 

3.  12^%. 

6.  i%. 

253*     III.  Ex.     Reduce  5  per  cent,  to  its  lowest  terms. 

Examples. 
Reduce  the  following  rates  to  their  lowest  terms :  — 

1.  90%.  Ans.j\,         5.  12^%. 

2.  50%.  Ans,  ^.         6.  8^%. 

3.  75%.  7.  16§%. 

4.  40%.  8.  33^%. 


9.  125%. 

10.  621%. 

11.  87^%. 

12.  31^%. 


254L,     III.  Ex.     Reduce  g  to  a  per  cent.     (Art.  238.) 
8  )  3.00 


.d1^  =  SlWc,Ans. 


172 


PERCENTAGE. 


Examples. 
Reduce  the  following  fractions  to  a  per  cent. :  — 


1.  fo-     Ans.SOfo- 

4-  1-                     1 

7.  J. 

2.  |.     Ans.  62J^c- 

5.   3^. 

8.  lA- 

3.  A-. 

6.  W. 

2S5,     III.  Ex.     Find  the  complement  of  15  per  cent.,  t.  e., 
what  it  wants  of  being  100  per  cent. 

100%  —  15%  =85%,  ^715. 

Examples 
Find  the  complement  of  the  following  rates  :  — 


1.  92%,.  Ans.  8%. 

2.  51%.^?*s.  49%. 

3.  11%. 


4.  83^%.  7.  18|%. 

5.  87^%.  8.  56|%. 

6.  33^%. 

S50.     To  Find  any  Per  Cent,  of  a  Number. 
III.  Ex.     What  is  25%  of  76  bushels  of  grain  ? 

76  bu.  X  .25  =:  19  bu.,  Ans. 

or  25  ^  =1 ;  ^  of  76  bu.  =  19  bu.,  Ans. 
Hence  the 

Rule.     To  find  any  per  cent,  of  a  number :  Multiple/  hy  the 

rate  per  cent.,  expressed  decimally. 


Examples. 
What  is 

1.  8%  of  $800?        Ans.  $64. 

2.  5%  of  324.40?  ^ws.  16.22. 

3.  20%  of  375  men? 

Ans.  75  men. 

4.  71%  of  800  trees? 

Ans.  60  trees. 

5.  12%  of  78  bu.?  Ans.  9/^  bu. 

6.  J%  of  $14.40  ?  Ans.  $.108. 


7.  4^%  of  12£.  6s.?  Ans.  lO^s. 

8.  235%  of  $85?  Ans.%\^'dl. 


9.  5%  off?  Ans.^\. 

10.  4%  of  110%  of  8750? 
Ans.  $33, 

11.  ^%  of  $200.75? 

12.  4^%  of  1000  gall.? 

13.  10%  of49£.  7s.  6d.? 

14.  66§%  of8d.  5h.  36  m.? 

15.  I  <fo  of  3  cwt.  2  qr.  (in  lbs  ) 

16.  75%  of8i? 


PERCENTAGE.  173 

17.  What  is  25%  of  125%  of  75%  of  50%  of  384  inches? 

18.  Farmer  F  kept  75  sheep  last  year,  and  sold  the  wool  for 
40  cents  per  pound.  He  has  20%  more  sheep  this  year,  and 
hopes  to  sell  his  wool  80%  higher.  How  many  sheep  has  he? 
What  does  he  hope  to  get  per  pound  for  wool  ? 

19.  If  12^%  of  $97.50  be  lost,  what  will  remain? 

20.  The  owner  of  a  field  of  wheat  allows  8-^%  of  the  wheat  for 
harvesting ;  what  will  be  the  owner's  share  if  80  bushels  are 
harvested  ? 

21.  I  had  $125  in  bills  on  the  Cochituate  Bank  when  it  failed. 
I  received  50%  of  this  in  good  money  from  the  bank,  and  after- 
wards 25%  of  the  remainder;  what  did  I  lose  ? 

22.  A  and  B  had  each  $2800  bequeathed  to  them.  A  gained 
15%  on  his  bequest,  and  B  lost  12^%  of  his  bequest.  How  much 
had  A  and  B  then  ? 

23*  Carefully  compiled  statistics  show  that  in  1860  in  the 
United  States  -1%  of  the  population  died  from  intemperance,  that 
1%  were  sent  to  prisons  and  almshouses,  that  §%  were  made 
orphans,  that  .001%  were  murdered,  and  .001^%  committed 
suicide,  from  the  same  cause.  Considering  the  population  to  have 
been  30000000,  what  was  the  number  of  victims  in  each  case  ? 
—in  all  ?  Ans.  380,700  in  all. 

24.  A  rope  12  yds.  long  shrank  2^%  on  being  wet.  Required 
its  length  after  shrinking. 

25.  If  a  yard  of  cloth  shrinks  6%  in  length  in  sponging,  what 
part  of  a  yard  in  length  will  it  be  after  sponging  ? 

26.  If  the  cloth  is  a  yard  wide,  and  shrinks  6%  in  length  and 
6%  in  width,  what  will  a  yard  contain  after  shrinking? 

-^^s.  Iiai  yd. 

27.  What  would  a  yard  have  contained  if  it  had  been  1|-  yds. 
wide  ? 

28!  How  much  cloth  originally  1^  yds.  wide  will  be  required 
to  make  a  suit  of  clothes  containing  9  full  sq.  yds.,  if  the  cloth 
purchased  shrinks  in  sponging  10%  each  way  ?         Ans.  8^  yds. 

^=*  For  Dictation  Exercises,  see  Key, 


J  74  PERCENTAGE. 

257.      To    FIND    100%  KNOWING   A    CERTAIN    OtHER    %. 

III.  Ex.     $25  is  624%  of  what  sura  ? 

If  $25  is  62^%  of  some  number,  1%  is  -^-^^  of  $25,  and  100%  is 
100  X  ^i  of  $25  =:  $40,  Ans.     Or 

62^%  =  |.  If  $25  is  |,  i  is  ^  of  $25,  and  |  is  8  X  i  of  $25,  &c. 
Hence  the 

Rule.  To  find  100%  knowing  a  certain  other  %:  Divide  the 
given  sum  hy  the  given  number  of  %,  and  multiply  by  100. 

Examples. 

1.  $31.35  is  5%  of  what  sura?  Ans.  $627. 

2.  $14.04  is  12%  of  what  sum?  Ans.  $117. 

3.  153000  raen  is  9%  of  what  number?  Ans.  1700000. 

4.  381  i^y  miles  is  11%  of  what  number?  Ans,  3470  miles. 

5.  250  is  4i%  of  what?  Ans.  6000. 

6.  I  is  15%  of  what?  Ans.  2|f. 

7.  84£.  14s.  is  87^%  of  what  sum?  Ans.  96£.  16s. 

8.  75  is  37^%  of  what  sum? 

9.  $700  is  140%  of  what  sura?  Ans.  $500. 

10.  The  sum  of  the  ages  of  a  father  and  son  is  44  years,  the 
eon's  age  being  10%  of  the  father's ;  what  is  the  age  of  each  ? 

Note 44  years  =  the  father's  age,  and  10^  more,  =  11-2-  of  the  father's 

age ;  .-.  1<%  =  yl^  of  44,  and  100^  =  100  X  y^^  of  44  years  =  40  years, 
father's  age. 

11.  Having  lost  12J^%  of  my  money,  I  have  $84  remaining; 
what  had  I  at  first  ? 

Note.  — The  complement  of  12^^  is  87i^=|;  .-.  8  X  |  of  84=* 
$96,  Ans. 

12.  1865  bushels  is  25%  more  than  what  number? 

13.  £67.76  is  12^%  less  than  what? 

14.  $4.14  is  3^%  more  than  what? 

15.  A  bankrupt  is  allowed  to  cancel  all  his  debts  by  paying  40 
cents  on  the  dollar ;  what  did  he  owe  to  a  person  to  whom  he 
paid  $2000.20  ? 

16.  An  attorney  receives  $1.26  for  collecting  a  bill  which  is 
J  %  of  the  bill ;  what  was  the  amount  of  the  bill  ? 


PERCENTAGE.  175 

17.  Drew  out  25^  of  my  deposit  in  a  bank;  of  this  I  have 
spent  $500,  which  is  4%  of  what  I  drew  out;  what  have  I 
remaining  in  the  bank?  Ans.  $37,500. 

18.  A  gain  of  101^^3%  in  the  population  of  the  United  States 
from  1830  to  18G0,  shows  an  increase  of  21000000 ;  what  was 
the  population  in  1830?  in  1860? 

A71S.  13000000  in  1830 ;  34000000  in  1860. 

19.  8%  allowance  is  given  a  debtor  for  making  present  pay- 
ment of  a  debt  due  at  a  future  time  without  interest ;  the  amount 
paid  is  $322.575 ;  what  was  the  sum  due?  Ans,  $350,625. 

J^"  For  Dictation  Exercises,  see  Key. 

S58.      To     FIND     WHAT    PeR    CeNT.    ONE    NuMBER    IS    OF 

Another. 
III.  Ex.     What  per  cent,  of  $50  is  $15? 
$15  is  ^f  of  $50  ;  |f,  reduced,  equals  30^^,  Ans.     Hence  the 
Rule.     To  find  what  %  one  number  is  of  another:    Divide 
iJie  number  expressing  the  part  by  the  number  with  which  it  is 
compared^  continuing  the  division  to  the  hundredths'  place. 

Examples. 

1.  $23  is  what  %  of  $92?  Ans.  "Ihofo^ 

2.  $15  is  what  %  of  $80?  Ans.  lS^<fo. 

3.  What  %  of  $18  is  2  cts.?  Ans.  |-%. 

4.  What  %  of  10  d-  is  2  w.  1  d.  Ans.  150%. 

5.  5  oz.  is  what  %  of  4oz.  7  pwt.  12 gr.?  Ans.  114f  % 

6.  4i-  is  what  %  of  15  ?  Ans.  30%. 

7.  What  %  of  48  doz.  is  6  doz.  ? 

8.  What  %  of  1600  men  are  1000  men? 

9.  What  %  of  1  dr.  is  1|  oz.  indigo? 

10.  What  %  of  1  doz.  is  one  score? 

11.  What  %  of  l£is  Is. 

12.  From  a  cask  containing  120  gal.  of  oil,  6  gal.  leaked  out; 
what  %  was  lost? 

13.  A  field  which  yielded  90  bu.  of  rye,  last  year,  yields  126 
bu.  this  year ;  what  is  the  gain  %  ? 

Note.      126  —  90  =  36,  the  gain  ;  f  §  =  40^,  Ans, 


176  PERCENTAGE. 

14.  If  gold  and  silver  coin  have  9  parts  pure  metal  to  1  part 
alloy  what  %  of  the  coin  is  alloy?  Ans.  10%.. 

15.  In  a  class  of  60  pupils  5  errors  were  made  in  spelling  1 
word  each;  required  the  %  of  errors.  Ans.  8-^%. 

16.  Subsequently,  in  spelling  10  woi-ds  each,  upon  the  slate, 
the  same  class  made  33  errors ;  required  the  %  of  correct  spell- 
ing. Ans.^4^%. 

17.  Of  a  certain  fsxrm  18  acres  are  pasture,  30  acres  wood- 
land, 12  A.  1  R.  acorn  field,  70  A.  a  wheat  field,  6  A.  2  R.  a 
potato  field,  and  3  A.  1  R.  a  garden ;  what  %  of  the  whole  farm 
is  each  part  ? 

18.  If  wood,  which  should  be  cut  4  ft.  in  length,  falls  short  2 
in.,  what  ^  should  be  deducted  from  the  price  ? 

1^^  For  Dictation  Exercises,  see  Key. 

S^9*    Miscellaneous  Examples  involving  Profit  and 

Loss. 

1.  A  lot  of  coal  cost  $7.50  a  ton ;  for  what  must  it  be  sold  to 
gain  33^%  ?  Ans.  $10. 

2.  Bought  40  reams  of  paper  at  $2  a  ream  j  at  what  price 
per  quire  must  I  sell  it  to  gain  20  ^  on  the  cost  ? 

Ans.  12  cents. 

3.  What  must  I  ask  apiece  for  lamps  that  cost  $4  a  doz.,  that 
I  may  make  25%  ?  Ans.  41  f  cts. 

4.  Sold  nutmegs  at  40  cents  a  pound,  and  lost  20  fo  ;  what  did 
they  cost  per  lb.  ?  Ans.  50  cents. 

5.  Sold  a  carriage  for  $240,  which  was  40%  less  than  it  cost ; 
required  the  cost.  Ans.  $400. 

6.  Lost  $15  by  selling  a  watch  at  25%  below  cost;  what  was 
the  cost?  Ans.  $60. 

7.  By  selling  a  lot  of  goods  for  $27.60, 1  gain  15%  ;  wl^t  did 
I  give  for  them  ?  Ans.  $24. 

8.  If,  by  selling  gloves  at  60  cents  a  pair,  20%  is  gained,  what 
was  the  cost  per  dozen  pairs  ?  Ans.  $6. 

fl.  What  would  have  been  the  cost  per  doz.  if,  by  selling  them 
at  60  cents,  6^%,  had  been  lost  ?  Ans.  $7.6a 


PROFIT  AND  LOSS.  177 

10.  What  was  my  property  worth  10  years  ago,  if  it  has  since 
increased  100%,  and  it  is  now  worth  $7000  ? 

11.  What  must  be  the  amount  of  my  sales  for  a  year,  that  1 
may  clear  $800  at  a  profit  of  16%  ? 

12.  If  I  pay  45  cents  a  pound  for  tea,  and  sell  it  at  56  cents, 
what%  do  I  gain? 

56  —  45  =  11,  the  gain  on  45 ;  ^  reduced  =:  24^%,  Ans, 

13.  What  %  is  gained  by  selling  goods  at  10  cents  a  yd.  which 
cost  8  cents  ?  Ans.  25  %. 

14.  What  %  will  be  lost  by  selling  a  book  for  75  cents  which 
cost  80  cents?  Ans.  6|%. 

15.  If  I  buy  a  horse  for  $75,  and  sell  him  for  $120,  what  is 
the  %  of  gain  ? 

16.  If  $1000  be  paid  for  a  lot  of  goods,  $640  be  received  for 
one  half  of  them,  and  $300  for  the  remainder,  is  there  a  gain  or 
loss,  and  what  %  ? 

17.  Bought  paper  at  $1.75  per  ream,  and  sold  it  at  20  cents 
per  quire  ;  what  %  did  I  gain  ? 

18.  What  %  should  I  gain  by  selling  at  1  cent  a  sheet? 

19.  Bought  150  beeves  at  the  rate  of  $42.50  each,  and  300 
sheep  at  the  rate  of  $4.50.  I  sold  the  lot  for  $10300 ;  what  did 
I  gain  %  ? 

20.  By  selling  wood  at  $6.50  per  cord,  I  gain  30%  ;  what  did 
I  give  per  cord  ? 

21.  Bought  a  cord  of  wood  for  $5,  and  sold  2  cd.  ft.  for  $1.62^; 
what  was  the  gain  %  ? 

22.  If  a  grocer  buy  15  cwt.  3  qr.  20  lb.  of  coffee  at  $9  per 
cwt.,  what  would  he  gain  or  lose  %  by  selling  the  lot  for 
8207.35? 

23.  What  %  is  lost  by  selling  a  lot  of  goods  for  f  of  their 
cost? 

24.  Wluit  %  is  gained  by  selling  a  lot  of  goods  for  2  times 
their  cost? 

25.  What  was  the  cost  of  a  lot  of  land  which,  selling  at  20% 
below  cost,  brings  $240  ? 

12 


178  INTEREST. 

26.  "What  was  the  original  value  of  a  share  in  a  bridge,  which, 
selling  at  35%  more  than  it  cost,  brings  $780? 

27.  Bought  a  cow  for  $87.50,  which  was  16f  %  more  than  her 
real  worth ;  what  was  her  worth  ? 

28.  A  grocer,  after  losing  11%  of  his  apples,  has  133.5  bbls. 
of  apples  left ;  if  they  cost  him  $2.50  per  bbl.  for  what  must 
they  be  sold  per  bbl.  that  he  may  lose  nothing  upon  his  purchase  ? 

29.  Bought  and  sold  250  lbs.  of  fish  and  gained  $3.75,  which 
was  42  f%  of  the  cost,  what  did  the  whole  cost,  and  what  did 
they  bring  per  lb.  ? 

Note.  —  Further  examples  in  Profit  and  Loss  will  be  found  in  Miscel- 
laneous Examples  in  Percentage. 


INTEREST. 

300.  Interest  is  a  certain  per  cent,  of  a  sura  of  money, 
paid  by  the  borrower  to  the  lender  for  its  use. 

The  Principal  is  the  money  lent. 

The  Amount  is  the  sum  of  the  Principal  and  Interest. 

The  Legal  Rate  is  the  rate  per  cent,  per  annum  established  by 
law. 

Usury  consists  in  taking  more  than  the  legal  rate. 

Note.  —  The  laws  often  impose  heavy  penalties  for  usury. 

361.  The  legal  rate  is  5%  'per  annum  in  England,  France, 
and  Louisiana;  7%  in  New  York  and  several  of  the  Western 
and  Southern  States  ;  8  %  in  the  Gulf  States,  excepting  La. ; 
6%  in  a  majority  of  the  United  States,  including  all  of  N.  Eng- 
land, and  in  Ireland,  Canada  and  Nova  Scotia. 

In  many  of  the  States  higher  rates  may  be  received  by  agree- 
ment ;  in  California,  any  rate. 

Note  I.  —  In  this  book,  ^%  will  be  understood  where  no  %  is  named. 

Note  II.  —  Business  men  reject  mills  from  the  products  when  less  than 
6,  and  call  5  or  more  1  cent.    In  this  book  mills  are  retained. 


METHODS  OF  COMPUTING  INTEREST.  179 

36^»     First  Method  of  Computing  Interest. 

At  G%,the  interest  on  $1  for  1  year,  or  12  months,  is  6  cents. 
If  for  12  months  the  interest  is  6  cents,  for  every  2  months  it 
is  1  cent ;  for  every  6  days,  which  is  j^  ^^  ^  months,  the  interest 
is  ^  of  1  cent  =  1  mill.  The  interest  will  be  in  the  same  pro- 
portion for  a  longer  or  a  shorter  period  of  time,  and  for  larger  or 
smaller  sums. 

III.  Ex.     What  is  the  interest  of  $200  for  5  y.  7  m.  19  d.  ? 

Operation. 

The  interest  of  $1     for  5  y.  =z5  X  $-06    ==    $.30 

"     "       «    7  m.  =3^  X     .01     =:      .035 

"       "     "       "    19  d.  =3^  X    .001=      .0031^ 

"        "     "       "    5y.7m.  19d.z=:  ^-^ 

«  «      «   $200  "        "  "      =  200  X  .338i=:$67.633^,  Ans, 

Hence  the 

Rule.  To  find  the  interest  on  $1  for  any  time  at  6%  :  Take 
6  times  as  many  eents  as  there  are  years^  one  half  as  many  cents 
as  there  are  months,  and  one  sixth  as  many  mills  as  there  are  days 
given,  and  find  their  sum. 

To  find  the  interest  on  any  number  of  dollars :  Multiply  the 
principal  by  the  interest  of$l  for  the  given  time. 

Examples. 
Find  the  interest  of  $1  for  the  following  times  :  — 


5.  1  y.  1  m.  10  d. 

6.  The  amount  of  $1  for  1  y. 

8  m.  Ans.  $1.10. 


1.  ly.  3  m.  6d.    Ans.  t07Q. 

2.  4y.  16d.         Ans.t2^2%. 

3.  4  m.  5  d.         Ans.  $.020^. 

4.  Im.  25d.        Ans.  $.009^.1 

7.  What  is  the  interest  of  |1  for  16  y.  8  m.  ?  Ans.  $1. 

8.  What  is  the  interest  of  $300  for  2  y.  5  m.  ?     Ans.  $43.60. 

9.  What  is  the  interest  of  $4.20  for  3  y.  6  m.  12  d.  ? 
10.  What  is  the  amount  of  $1000  for  7  y.  10  m.  18  d.  ? 

Ans.  $1473, 


180  PERCENTAGE. 

303»     Second  Method  op  Computing  Interest. 
"We  see  by  Ex.  7,  Art.  2G2,  that  the  interest  of  $1  for  I67. 
8  m.,  or  200  monthsj  is  the  same  as  the  principal ;  this  is  true  of 
any  sum   at  6%.     Upon  this  fact  is   based  an  ingenious  and 
practical  method  of  computing  interest. 

The  following  table  shows  the  relation  which  the  interest  bears 
to  the  principal  at  6^  for  various  periods  of  time :  — 

Table. 


Interest  for  16 

y.  8  m.           or 

200  m. 

znz 

PrincipaL 

(( 

u 

8 

y.  4  m.            or 

100  m. 

z=z 

i  do. 

u 

tl 

4 

y.  2  m.           or 

50  m. 

— 

i  do. 

u 

It 

2 

y.  1  m.            or 

25  m. 

zzz 

i  do. 

It 

u 

1 

y.  15  d.           or 

121m. 

■^iz 

tV^o. 

u 

11 

5 

y.  6  m.  20  d.  or 

66|m. 

z=z 

^do. 

tl 

It 

2 

y.  9  m.  10  d.  or 

33|m. 

.z=z 

ido. 

u 

tt 

1 

y.  4  m.  20  d.  or 

16|m. 

z= 

tV^o. 

u 

ii 

8  m.  10  d.  or 

8^m. 

=r 

^do. 

(( 

11 

3 

y.  4  m.            or 

40  m. 

— 

ido. 

i( 

u 

1 

y.  1  m.  10  d.  or 

13|-m. 

=rr 

J  of  1  do. 

tl 

11 

1 

y.  8  m.           or 

20  m. 

— 

tV  do- 

u 

tl 

6  m.  20  d.  or 

6|m. 

rzr 

i  of  J^  do. 

u 

11 

3  m.  10  d.  or 

3|m. 

— 

i  of  3-V  do. 

tl 

u. 

10  m. 

— 

i^do. 

11 

tt 

5  ra. 

r=: 

i  of  ^\  do. 

It 

tt 

2  m.  or  60  d. 

— 

ikdo. 

tl 

It 

1  m.  or  30  d. 

: — 

^of^i^do. 

11 

tt 

1  m.  or  15  d. 

r= 

iof^l^do. 

tl 

tt 

1  m.  or  12  d. 

m: 

iof^l^do. 

t& 

tt 

1  m.  or  10  d. 

r=r 

1  of  ^J^  do. 

tl 

ii 

^  m.  or    6  d. 

=  tVo'' 

10  0  or  loWdo. 

tl 

It 

-:^Q  m.  or    3  d. 

i  of  j-,\^  do. 

tl 

tt 

J5  m.  or    2  d. 

=: 

i  0^  ToW  do. 

(( 

tl 

■^Q  m.  or    1  d. 

=: 

1  of  T^  do. 

To  obtain  the  interest  at  6%  by  this  method  for  200  moni^hs, 
20  -months,  60  days,  or  6  days,  nothing  is  required  but  remov- 
ing ike  decimal  point ;  facility  in  computing  for  any  other  period 
of  time,  depends  upon  suhdividing  the  given  time  into  convenient 
factors  and  multiples  of  200  months^  20  months,  60  days,  or  6 
days^ 


SIMPLE  INTEREST. 


181 


III.  Ex.,  I.     What  is  the  interest  of  1480  for  1  y.  3  m.  ? 


1st  Method. 
$480  multiplied  by 
.075  int.  of  $1  for  1  y.  3  m. 

2400 
3360 


$36,000,  Arts. 


2d  Method. 
Principal,  (1), 

Int.  for  10  m.  -^^  of  (1)  =(2),      24 
"      "     5  m.   |of(2)=z(3),      12 

"     "    ly.  3  m.  =zAns.$3Q 


III.  Ex.,  II.     Required  the  amount  of  $872.32  for  6  y.  2  m. 

6d. 

2d  Method. 

Principal                              (1),  $872.32 

Int.  for  50  m.      l  of  (1)  =  (2),  218.08 

20  m.    ^Lof  (l)i=(3),  87.232 

4  m.      ^of(3)=:(4),  17.446 

6d.^J^of(l)::r:(5),  .872 


1st  Method. 
$872.32  mult'd  by 
1.371am'tof$lfor 

6  y.  2  m.  6  d. 

87322 
610624 
261696 
87232 


$1195.95072,  Ans. 


6y.  2  m.  6d.    ^?i5.  $1195.950. 


III.  Ex.,  III.     Find  the  interest  of  $762.75  for  3  y.  10  m. 

29  d. 

1st  Method. 

762.75,  multiplied  by 


.234|,  int.  of 


for  3  y.  10  m.  29  d. 


63562^ 
305100 

228825 
152550 

Ans   $179,119+  int.  of  $762.75  for  3  y.  10  m.  29  d. 
2d  Method. 
Principal  (1),    $762.75 


Int.  for  40  m.  i                 (1)  =  (2), 
"     "     6f  m.lofJ^of(l)=:(2), 
"     "6  d.    ^1^           (1)  =  (4), 
«     "3d.        ^             (4)  ==(5), 

152.55 
25.425 
.762 
.381 

''    "    3y.  10  m.  29d.              ^n5.  $179. 118- 

182  PERCENTAGE. 

Examples. 

1.  Find  the  interest  of  |100  for  1  y.  4  m.  Ans.  |8. 

2.  $75,085  for  1  y.  8  m.  6  d.  Ans.  $7,583. 

3.  $987.35  for  4  y.  2  m.  28  d. 

4.  136.18  for  3  m.  7  d. 

5.  $96.34  for  1  m.  10  d. 

6.  $130.50  for  2  y.  9  m.  13  d. 

7.  $800.20  for  3  y.  4  m.  12  d. 

8.  $16.82  for  9  m.  27  d. 

9.  $1000  for  3  y.  10  m.  2  d. 

10.  $25.50  for  1  y.  1  m.  1  d. 

11.  Find  the  amount  of  $14.98  for  2  y.  6  m.  29  d. 

Ans.  $17,299. 

12.  Find  the  amount  of  $490.82  for  4  y.  7  m.  17  d. 

13.  Find  the  amount  of  $97.65  for  5  y.  11  m.  14  d. 

264:*    To  FIND  Interest  at  any  other  Rate  than  Q^fc 

III.  Ex.     What  is  the  interest  of  $490  for  1  y.  5  m.  24  d. 

at7%? 

Principal,  (1),$490. 


Int.  for  1  yr.,  .07  X  (1)  =  (2),      34.30 

«  "    4  m.,  ^  of  (2)      z=(3),      11.433-1- 

«  "    1  m.,  \  of  (3)      =  (4),        2.858-1- 

«  "    24  d.  i  of  (3)*    z=z  (5),        2.286-}- 

«  "    1  y.  5  m.  24  d.,   Ans.,    $  50.87 7-|- 

RuLE  I.  To  find  interest  at  any  %  :  Miid  the  interest  for  1 
year  by  multiplying  the  principal  by  the  given  rate  ;  and  from  that 
interest  compute  the  interest  for  the  given  time,  by  Practice.  (Art. 
i49.)     Or, 

Rule  II.  Find  the  interest  at  Q^c^  «^^  increase  or  diminish 
that  interest  as  the  given  %  is  greater  or  less  than  6%. 

Thus,  for  7  %  take  7  times  ^  of  the  interest  at  6  % ,  or  add  | ; 
for  5  %  take  5  times  |  of  the  interest  at  6  % ,  or  subtract  -^ ;  for 
7^  %  take  7^  times  I  of  the  interest  at  6  %,  or  add  |,  &c. 

*  24  d.  =  f  of  1  mo.  =  -I  of  4  mo. 


SIMPLE  INTEREST.  183 

Examples. 
Find  the 

1.  Interest  of  $837.36  for  3  y.  2  mo.  at  7%. 

Ans.  $185,614+. 

2.  Interest  of  $400.08  for  2  y.  4  m.  2  d.  at  9%. 

Ans.  $84,216+. 

3.  Amount  of  $640  for  1  y.  6  m.  at  7%.         Aiis,  $707.20+. 

4.  Interest  of  $75.85  for  10  m.  3  d.  at  9%.       Ans,  $5,745+. 

5.  Amount  of  $416  for  3  y.  16  d.  at  7%. 

6.  Interest  of  $450  for  5  y.  4  m.  3  d.  at  8%. 

7.  Interest  of  $658  for  9  m.  at  ^%.  Ans.  $2,467+. 

8.  Amount  of  $325  for  3  d.  at  7^%.  Ans.  $325,196+ 

9.  Interest  of  $896  for  2  y.  6^  m.  at  6|%.   Ans.  $151,822+ 

10.  Interest  of  $187.50  for  2  m.  12  d.  at  10%. 

2^5*.    To  FIND  Interest  on  English  Currency. 

III.  Ex.     Find  the  interest  of  10  £.  15  s.  6  d.  for  16  yr.  10 

mo.  at  6%. 

Operation. 

10  £.  15  s.  6  d.rr:  10.775 £. 
10.775  £  multiplied  by 
1.01,  interest  of  1  £  for  16  y.  10  m. 

10.88275  £  =  10  £.  17  s.  7  d.  3+  qr.,  Ans, 
Hence  the 

Rule.  Reduce  the  shillings,  pence,  and  farthings  to  the  deci- 
mal  q/*  1  £  t  (Art.  244),  compute  interest  as  in  Federal  Money, 
and  reduce  the  decimal  of  1  £  to  s.  d.,  S^c,  (Art.  245.) 

t  Shillings,  pence,  and  farthings  may  be  reduced  to  the  decimal  of  a 
t^ound  by  inspection  as  follows  : 

iLLUSTRATiox.  C'a^/  half  the  number  of  even  shillings  tenths^ 

5  s.  10  d.  3  far.  =  .296  £,    and  the  odd  shilling,  if  any,  5  hundredths.    Re- 
5  s.  ^  .25  (iuce  pence  and  farthings  to  farthings,  and  call 

10  d.  3  f.  =  43  f.  =  .045        them  thousandths,  adding  one  to  the  number 
.•.5  s.  10  d.  3  f.  =  .295  £.  when  it  exceeds  12,  and  2  when  it  exceeds  36. 
The  decimal  of  a  pound  may  be  reduced  to  shillings,  pence,  and  far- 
things, as  follows : 


184  PERCENTAGE. 

Examples. 
Find  the 

1.  Interest  of  10  £.  15  s.  for  2  y.  9  m.  10  d. 

Ans.  1£.  15  s.  10  a. 

2.  Interest  of  17  s.  for  1  y.  1  m.  4  d. 

3.  Interest  of  241  £.  10  s.  6  d.  for  1  y. 

4.  Interest  of  15  £.  7  s.  10  d.  for  3  y.  11  m.  14  d.  at  4%. 

5.  Interest  of  27  £.  7  s.  11  d.  for  1  y.  7  m.  18  d.  at  5%. 

6.  Amount  of  20  £.  8  s.  for  4  y.  2  m.  27  d.  at  8%. 

7.  Interest  of  482  £.  10  s.  for  3  y.  2  m.  at  a  rate  equal  to  1 
shilling  on  £1.  Ans.  76  £.  7  s.  11  d. 

^0G»  In  computing  interest,  it  is  often  necessary  to  find  the 
time  between  two  dates.  This  may  be  done  by  subtraction,  as  in 
Art.  204,  or  mentally  as  follows : — 

III.  Ex.,  I.  What  is  the  time  from  May  19,  1860,  to  Mar. 
28,  1862  ? 

From  May  19,  '60,  to  May  19,  '61,  =  1  y. 
From  May  19,  '^1,  to  Mar.  19,  '62,  =         10  m. 

From  Mar.  19,  '62,  to  Mar.  28,  '62,  z=: 9  d. 

From  May  19,  '60,  to  Mar.  28,  '62,  =  1  y.  10  m.  9  d.,  Ans. 

III.  Ex.,  II.  What  is  the  time  from  May  19,  '60,  to  Mar. 
15, '62? 

From  May  19,  '60,  to  Mar.  19,  '62,  as  above,  =  1  y.  10  m. ;  but  to 
Mar.  15,  '62,  it  being  4  days  less,  it  is  1  y.  9  m.  26  d.     Hence 

Rule  I.  Find  the  number  of  years  and  months  between  the  first 
date  and  the  same  day  of  the  month  in  the  second  date.  Jf  this 
falls  short  of  the  true  time,  add  the  difference  of  days  ;  if  it 
reaches  beyond  it,  subtract  the  difference  of  days. 

Rule  II.  Find  the  number  of  years  and  entire  calendar 
months  between  the  dates,  and  then  the  remaining  days.  (p.  329.) 

Note.  —  The  answers  given  in  the  book  are  by  Rule  I.  Answers 
obtained  by  both  rules  are  given  in  the  Key. 

Illustration.  Call  every  tenth  2  shillings,  and  every  5 

.584  £  =  11  s.  8  d.  1  far.  hundredths,  1  shilling.     Call  the  remainder 

.55    £  =  lis.  farthings,  subtracting  1  if  the  mmiber  ex- 

.034  £  =  33  far.  =  8  d.  1  far.   ceeds  12,  and  2  if  it  exceeds  36. 
.'.  .584  £:=  lis.  8cl.  Ifar.  ^ 


SIMPLE   INTEREST.  185 

It  is  just  to  compute  interest  by  the  exact  number  of  days  in  each  cal* 
endar  month.  This  is  the  method  employed  by  the  United  States  Gov- 
ernment and  throughout  Great  Britain.  For  method  of  computing  inter- 
est at  7yY/o,  see  Appendix,  p.  335. 

367.     Miscellaneous  Examples. 
What  is  the  interest  of 

1.  $270.87  from  Oct.  17,  1860  to  Dec.  28,  1863.? 

Ans.  S51.961+, 

2.  $400.37  from  Mar.  14,  1857,  to  Sept.  9,  1859? 

Ans.  $59,722^-. 

3.  11000  from  Nov.  11,  1856,  to  Aug.  15,  1862,  at  7%  ? 

Ans.  $403,277+. 

4.  $19.80  from  Oct.  15,  1859,  to  Apr.  19,  1860,  at  5%? 

Ans.  $0,506. 

5.  $130.16  from  Feb.  7,  1866,  to  Dec.  1,  1870,  at  8%  ? 

Ans.  $50,154+. 

6.  $99.99  from  Jan.  15,  1860,  to  Mar.  10,  1863,  at  3%  ? 

Ans.  $9,457+. 

7.  $62.50  from  Aug.  3,  1862,  to  Apr.  11,  1863,  at  7^%  ? 

Ans.  $3,229+ 

8.  $175  from  Dec.  4,  1861,  to  May  1,  1864? 

9.  $2000  from  Sept.  8,  1859,  to  Jan.  3,  1861  ? 

10.  $120.90  from  July  10,  1865,  to  Feb.  2,  1868,  at  9%  ? 

11.  $456.82  from  June  15,  1830,  to  June  6,  1865,  at  1%  ? 
Find  the 

12.  Amount  of  $365  from  Dec.  12,  1860,  to  Mar.  7,  1861. 

Ans.  $370,171—. 

13.  Amount  of  $58.80  from  Nov.  1,  1844,  to  Feb.  1,  1849, 
at  7^%. 

14.  Interest  of  $40.75  from  Aug.  19,  1835,  to  June  17,  1838, 
at  4%. 

15.  Interest  of  $150  from  July  5,  1860,  to  Mar.  17,  1862. 

16.  Interest  of  £1000  from  July  8,  1858,  to  Mar.  5,  1860,  at 

nr  Amount  of  430  £.  7  s.  8f  d.  from  July  15,  1870  to  Oct.  5, 
1874,  at  12%.  Ans.  648 £.  8 s.  11  d. 


186 


PERCENTAGE. 


IS*  Interest  of  15  £.  10  s.  from  Dec.  28  to  Jan.  4,  by  the  exact 
nu'^iber  of  days.  Ans.  4  d.  1  -)-  f. 

19.  Interest  of  $1600  from  Aug.  10,  1864,  to  Jan.  1,  1865,  by 
the  exact  number  of  days,  at  5  ^ .  Ans.  $32. 

20.  A  man  gave  his  note  (Art.  268)  May  7,  1830,  for  $1800, 
with  interest;  what  sum  would  discharge  the  note  June  21, 
1834?  Ans.  $2245.20. 

21.  In  settling  with  a  person,  Jan.  1,  1859,  I  found  I  owed 
him  $387.20 ;  for  this  sum  I  gave  my  note  on  interest  at  7  %  ; 
what  should  I  pay  to  discharge  this  note  Oct.  20,  1859  ? 

22.  There  are  two  notes,  one  dated  Jan.  19, 1850,  for  $375.83, 
the  other  dated  May  19,  1851,  for  $76.19  ;  what  is  the  amount 
of  both  notes  Jan.  1,  1852,  each  bearing  interest  from  its  date  ? 

23.  Reed  and  Prescott  bought  goods  to  the  following  amounts, 
agreeing  to  pay  7%  interest  from  the  date  of  purchase  :  July  8» 
1864,1470;  July  28,  $235;  Oct.  2,  $206.  What  will  be  the 
amount  due  Jan.  1,  1865  ?  A?is.  $937,366+. 

24.  lYhat  is  the  balance  of  the  following  account  Jan.  1,  1862, 
and  dae  to  whom,  reckoning  interest  on  each  item  from  its  date  ? 


Leonard  Harris 
Dr  In  %  with  Martin-  LiNCOLN.f 


Cr. 


1861. 

1861. 

— ' 

May     5 

To  Wool, 

$400 

00 

Feb.    16 

By  Oats, 

$200 

00 

:rune     7 

"  Goods, 

420 

00 

May     5 

"   Corn, 

174 

00 

Aug.    28 

"  Goods, 

225 

00 

Sept.   14 

"   Hay, 

380 

00 

Ans.  Due  M.  Lincoln,  $301,505. 

25.  Find  the  balance  due  Cabot  in  the  following  account,  Oct. 
1,  1865,  interest  at  6%,  from  the  date  of  the  items  :  — 

Arthur  Lee 
Dr.  In  %  with  Geo.  D.  Cabot.  Cr. 


1865. 
Mar.  29 
Apr.   22 


To  Mdse., 
"   Cash, 


$476  93 
869  82 


1865. 
Apr.  24 
May    15 


By  Mdse., 
"    Mdse., 


379 


t  The  Dr.  side  of  this  account  shows  what  goods  Harris  has  bought 
>f  Lincoln  ;  the  Cr.  side  what  he  has  sold  to  Lincoln. 


PARTIAL  PAYMENTS.  187 

PARTIAL    PAYMENTS. 

Notes. 

368*  A  Promissory  Note,  usually  called  a  Nbte^  is  a 
Ivritten  promise  to  pay  money  or  merchandise  for  value  re- 
ceived. 

S69.  The  gum  promised  is  called  the  Principal  or  Face 
of  the  Note,  and  should  be  written  in  words. 

370.  In  order  that  a  note  shall  draw  interest  from  date,  '^with 
interest "  must  be  inserted  in  it ;  but  notes  on  demand^  without 
interest  specified,  draw  interest  from  the  time  payment  is  de- 
manded ;  and  notes  on  time,  from  the  time  when  due,  if  not  then 
paid,  though  interest  is  not  specified. 

S71,  To  be  negotiable,  i.  e.,  transferable  or  salable,  a  note 
must  be  made  payable  "  to  order"  or  "bearer."  If  "  to  order," 
the  holder  cannot  transfer  it  without  endorsing  it,  i.  e.,  writing 
his  name  on  the  hack  of  it. 

The  endorser  of  a  note  becomes  liable  for  its  payment  under 
certain  circumstances.  He  may  endorse,  without  becoming  lia- 
ble, by  writing  above  his  name  "  without  recourse." 

272m  According  to  custom,  and,  in  many  States,  by  law,  a 
note  is  not  considered  due  till  three  days  after  the  time  specified 
for  its  payment.  These  are  called  days  of  grace,  and  interest 
is  taken  for  them. 

Note.  —  A  note  is  said  to  mature  when  it  becomes  due. 

273,  Partial  Payments  are  payments  in  part  of  notes  or 
other  obligations. 

374,  To  compute  the  interest  on  notes,  when  partial  pay- 
ments have  been  made,  observe  the  following,  called 

The  United  States  Rule. 

1.  Find  the  amount  of  the  sum  due  from  the  time  interest  com- 

mences  to  the  time  of  the  first  payment ;  subtract  the  payment,  if 

it  will  cancel  the  interest,  and  consider  the  remainder  a  new  prin- 

cipal ;  find  the  amount  of  this  new  principal  from  the  time  of  ths 


188  PERCENTAGE. 

Jlrst  payment  to  the  time  of  the  second  ;  subtract  the  second  pap'^ 
ment  as  before,  and  so  proceed  to  the  time  of  settling  the  note. 

2.  By  the  decisions  of  the  United  States  Court,  when  a  pay- 
ment  will  not  cancel  the  interest  due,  interest  is  computed  on  the 
principal  till  surfficient  sums  have  been  paid  to  cancel  all  the  in^ 
terest  due,  when  all  the  payments  are  subtracted  from  the  amount 
due  as  if  paid  in  one  sum. 

Note. —  When  partial  payments  are  made,  the  account  is  kept  by  en- 
tering the  same,  with  their  dates,  on  the  back  of  the  note.  The  entrioi 
are  called  endorsements. 

Examples. 

1.  Suppose  a  note  for  $1908.42,  dated  Aug.  9,  1851^10  be  on 
interest  till  Feb.  15, 1852,  when  a  payment  of  $1732.59  is  made; 
what  sum  will  remain  due  ?  Ans.  $234,991. 

2.  Suppose  the  above  balance  ($234,991)  to  remain  on  inter- 
est till  April  3,  1853,  when  another  payment  of  $50  is  made; 
what  will  then  be  due  ?  Ans.  $200.97. 

3.  Suppose  the  balance  ($200.97)  to  continue  on  interest  to 
Jan.  9,  1860,  what  will  be  due  at  that  time  ?     * 

4.  A  note  for  $75.83,  with  interest,  is  dated  Jan.  19,  1850; 
suppose  $15  to  be  paid  July  15,  1850,  what  will  remain  due  ? 

5.  Suppose  $40  of  the  above  balance  to  be  paid  April  13, 
1852,  what  will  then  be  due  ? 

6.  If  this  balance  remains  on  interest  till  Feb.  7,  1853,  what 
sum  will  then  be  due?  Ans.  $31,105+. 

7.  A  note  for  $50,  dated  Jan.  1,  1862,  is  on  interest  at  6% 
till  April  15,  1862,  when  a  payment  of  $25.87  is  made.  What 
sum  will  remain  due  ? 

8.  If  the  balance  of  the  above  should  remain  on  interest  at  7% 
till  Jan.  13,  1863,  what  will  then  have  to  be  paid  to  discharge 
the  note? 

Promissory  Note.     (Art.  268.) 

9.  $300.  Philadelphia,  April  5,  1857. 
On  demand,  I  promise  to  pay  E.  Varnum,  or  bearer,  three 

hundred  dollars,  with  interest,  value  received.  C.  J.  Potter, 


PARTIAL  PAYMENTS.  189 

On  the  above  note  were  the  following  endorsements :  — 
Eeceived,  May  29,  1860,  $217.49. 
"         Apr.  23,  1862,  $50. 
What  will  be  the  balance  due  on  the  above  note  December  15, 

1869?  Am.  $153,275. 

Operation. 

Principal, $300.       ,  on  interest  from  Apr.  5,  '51. 

Interest  on  principal,    .         56.70  ,  to  May  29,  '60,  (3  y.  1  m.  24  d.) 

Amount, $356.70 

let  payment,  ....       217.49 

2d  principal,  ....    $139.21  ,  on  interest  from  May  29,  '60. 

Interest  on  2d  principal,       15.869,  to  Apr.  23,  '62,  (1  y.  10  m.  24  d.) 

Amount, $155,079 

2d  payment,  ....         50. 

3d  principal,  ....     $105,079,  on  interest  from  Apr.  23,  '62. 
Interest  on  3d  principal,      48.196,  to  Dec.  15,  '69,  (7  y.  7  m.  22  d.) 
Amount, $153,275,^/15. 

10.  $1000.  Burlington,  Oct.  5,  1854. 
For  value  received,  I  promise  to  pay  to  the  order  of  Joseph 

P.  Battles,  one  thousand  dollars,  with  interest,  on  demand. 

J.  BUSNELL. 
ENDORSEMENTS. 

Received  of  within  Dec.  8,  1854,     $125. 

«         «       "      May  12,  1855,     316. 

«        <'       "      Sept.  2,  1855,      417. 

«         "       "      Mar.  9,  1856,       100. 

What  balance  remained  due  June  15, 1856?        Ans.  $93.353 -f  . 

11.  $700.  Lancaster,  April  5,  1847. 
On  demand,  with  interest  at  7%,  we  promise  to  pay  H.  K. 

Oliver,  or  order,  seven  hundred  dollars,  value  received. 

Warren  Burton  &  Co. 

ENDORSEMENTS. 

Received  of  within,  Oct.  29,  1850,  $217.49. 
«       «      July  23, 1852,    200.00. 
What  remained  due  Dec.  12,  1859  ?  ^/i*.  $814,681. 


190  PERCENTAGE. 

Note.  —  In  the  following  examples  in  Partial  Payments,  consider  each 
note  to  be  on  demand  with  interest  from  its  date,  unless  otherwise  specified. 

12.  A  note  for  $960  is  dated  Nov.  16,  1855,  on  which  was 
paid  $140,  Nov.  11,  1856;  $80,  July  30,  1857;  $70,  Jan.  2, 
1858 ;  and  $100,  Dec.  1,  1858.  What  balance  is  due  Oct.  30, 
1859,  reckoning  interest  at  7%  ?  Ans.  $806,077. 

13.  $350.  Bristol,  Jj!?rz7  5,  1850. 
For  value  received,  we  jointly  and  severally  promise  John 

Ingalls  to  pay  him,  or  order,  three  hundred  fifty  dollars,  on 
demand,  with  interest  at  5  ^  per  annum,  after  three  months  from 
date.  Hood  &  Bishop. 

On  this  note  were  the  following  endorsements:  Nov.  1,  1852, 
received  $87  ;  March  7,  1855,  received  $150  ;  Feb.  19,  1858, 
received  $115.   What  was  due  Sept.  15, 1862  ?     Ans.  $125.Gl-f . 

14.  A  note  for  $935  is  dated  Sept.  1, 1855,  on  which  was  paid 
$125.75,  Jan.  15,  1856  ;  $250,  March  25,  1861 ;  $300,  IMay  10, 
1861.     What  was  the  balance  due  July  1,  1861  ?  Ans.  1549.713. 

Opekation. 

Principal, $935.       ,  on  interest  from  Sept.  1,  1855. 

Interest  on  principal,    .  20.881,  to  Jan.  15,  '56  (4  m.  14  d.). 

A.mount, $955,881 

1st  payment,  ....  125.75 

2d  principal,  ....  $830,131,  on  interest  from  Jan.  15,  ^5Q. 

Interest  on  2d  principal,  258.724,  to  Mar.  25,  '61  (5  y.  2  m.  10  d.). 

Interest  on  2d  principal,  6.225,  to  May  10,  '61  (1  m.  15  d.*). 

Amount $1095.08 

2d  and  3d  payments,    .  550.00  ,  both  required  to  cancel  interest. 

3d  principal,    ....  545.08  ,  on  interest  from  May  10,  '61. 

Interest  on  3d  principal,  4.633,  to  July  1,  '61  (1  m.  21  d.). 

Amount, $549,713,  Ans. 

15.  $500.  Salem,  April  1,  1855. 
For  value  received,  I  promise  to  pay  W.  J.  Rolfe,  or  order, 

five  hundred  dollars,  on  demand,  with  interest  from  Oct.  1, 1855. 

Irenas  Edwards. 

♦  See  Art.  274,  Rule,  2d  Clause. 


PARTIAL  PAYMENTS.  191 

ENDORSEMENTS. 

Received  of  the  within,  Apr.  1,  1856,  $12. 
"  "    "         "      Apr.  1,  1857,  1100. 

«  "    «         "      Apr.  1,  1858,  1100. 

What  is  due  June  19,  1858?  Ans.  $363,646+. 

16.  A  note  for  |1000  is  dated  June  1,  1860.  The  endorse- 
ments are :  $75,  paid  Aug.  1, 1860  ;  $125.75,  paid  Dec.  15, 1860 ; 
$250,  paid  Feb.  25,  1866;  and  $300,  paid  Apr.  10, 1866.  What 
will  be  due  on  this  note  June  1,  1866?  Ans.  $549.713-f . 

17.  A  note  for  $790,  dated  Oct.  9,  1862,  is  endorsed  Sept.  6, 
1863,  with  $320;  Jan.  30,  1864,  with  $10;  Oct.  9,  1864,  with 
$190.     What  balance  is  due  Feb.  3,  1865,  interest  at  5%  ? 

Ans.  $338.77+. 

iSr  A  note  for  $800,  dated  Jan.  15,  1860,  is  on  interest  after 
6  months,  and  is  endorsed  Apr.  18,  1861,  $100;  Jan.  1,  1863, 
$70;  and  June  15, 1864,  $62.50,  —  interest  being  at  7%  .  What 
was  due  July  15,  1865?  Ans.  $830,415+ 

19!  Upon  a  note  of  $425,  dated  July  13,  1859,  there  are  the 
following  endorsements:  August  10,  1861,  $50;  November  18, 
1862,  $150.  What  will  be  due,  if  the  note  is  settled  July  13, 
1863? 

20!  A  note  for  $250,  dated  May  15,  1838,  is  endorsed  Feb. 
25,  1841,  $111.66§;  Oct.  19,  1842,  $15;  May  9,  1848,  $62; 
and  Oct.  15,  1849,  $100.30.  Required  the  balance  due  July  22, 
1851. 

21*.  A  note  for  $489  is  dated  Jan.  20,  1850,  and  endorsed  as 
follows:  June  26,  1850,  received  $50;  Feb.  26,  1852,  received 
$40;  July  8,  1855,  received  $90;  Jan.  26,  1856,  received  $200; 
June  20,  1856,  received  $200.  If  this  note  was  on  interest  from 
three  months  after  date,  what  was  due  Nov.  20,  1856? 

275*  The  following  rule  for  Partial  Payments  is  in  general 
use,  when  the  whole  period  of  time  is  less  than  one  year :  — 

Rule.  Fi7id  the  amount  of  the  principal  for  the  tvhole  time 
the  note  is  on  interest ;  find,  also,  the  amount  of  each  payment 
from  the  time  it  is  rnade  to  the  time  of  settling  the  note  ;  and  de~ 
duct  the  sum  of  the  payments^  with  their  interest,  from  the  amount 
of  the  principal. 


192  PERCENTAGE. 


III.  Ex.     1800.  Burlington,  July  7,  1860. 

Three  months  after  date,  I  promise  to  pay  John  Thetford, 
or  bearer,  eight  hundred  dollars,  with  interest. 

Benjamin  Stokes. 

On  the  back  of  the  above  note  were  recorded  the  following 
payments :  — 

Received,  Aug.  16,  1860,  $200. 
"  Oct.  8,  1860,  $480. 

«  Feb.  20,  1861,  $49.92. 

What   balance  was  due  at  the  time   of   settlement,  July  1, 
1861  ?  Ans.  $84.65. 

Operation. 
Principal,  $800,  from  July  7,  '60,  to  July  1,  '61  (11  m.  24  d.), 

amounts  to $847.20 

1st  pay't,  $200,  from  Aug.  16,  '60,  to  July  1,  '61 

(10  m.  15  d.),  amounts  to  .         .         .     $210.50 

2d  pay't,  $480,  from  Oct.  8,  '60,  to  July  1,  '61 

(8  m.  23  d.),  amounts  to  .        .        .       501.04 

3d  pay't,  $49.%2,  from  Feb.  20,  '61,  to  July  1,  '61 

(4  m.  11  d.),  amounts  to  ...         51.01  — 

762.55 


Balance  due,        .        .        .  Ans.    $84.65 


1.    $10000tV^.  Concord,  OcL  4,  1863. 

In  two  months  from  date,  I  promise  to  pay  to  the  order  of 
Benjamin  Tyler,  at  Suffolk  Bank,  Boston,  ten  thousand  -^^^^ 
dollars,  with  interest,  value  received. 

Thomas  Beeman. 

endorsements. 

Received  of  within,  $672.41,  Nov.  5,  1863. 
«         "       "        $7682.42,  Nov.  15,  1863. 
"        **       «        $437.98,  Nov.  16,  1863. 

"         "       "        $833.42,  Nov.  19,  1863. 

What  was  the  balance  due  on  the  above  note,  when  the  note 
became  due  ?  Am,  $443,555. 


PARTIAL  PAYMENTS.  193 


2.  $1200  Albany,  April  1,  18G2. 

One  year  from  date,  for  value  received,  I  promise  to  pay  to 
J.  V.  Smiley  or  order,  twelve  hundred  dollars,  with  interest,  at 
7^.  Okrin  Jones. 

The  above  was  indorsed  as  follows :  — 

April  11,  1862,  $161.08;  July  18,  1862,  $224.14; 
July  27,  1862,  $17.90;  Jan,  28,  1863,  $100,25. 

What  was  still  due  April  1,  1863? 

Ans,  $756,565. 

Q76.     The  following  is  the 

Connecticut  Rule. 

1.  When  a  year's  interest  or  more  has  accrued  at  the  time  of 
a  payment,  and  always  in  case  of  the  last  payment,  foUow  the 
Governmeivt  Eule,    (Art,  274.) 

2.  When  less  than  a  year's  interest  has  accrued  at  the  time  of 
a  payment,  except  it  be  the  last  payment,  find  the  difference  between 
the  amount  of  the  principal  for  ■an  entire  year,  and  the  amount 
of  the  payment  for  the  balance  of  a  year  after  it  is  made  ;  this 
difference  will  form  the  new  prin-cipal. 

3.  If  the  interest  which  has  arisen  at  the  time  of  a  payment 
exceeds  the  payment,  no  interest  will  be  'Computed  upon  the  pay- 
ment,  but  only  up&n  the  principal. 


1,  $1000,  Hartford,  March  9,  1855. 

In  one  year  from  date,  for  value  received,  I  promise  to  pay 
Geo.  Yates  or  order,  one  thousand  dollars,  with  interest,  at 
6^.  Joseph  W,  Boomer,  Jr. 


ENDORSEMENTS, 

Received  Nov.  19,  1855,  $204;  Mar.  3,  1857,  |50; 
June  15,  1858^  $600  ;  Nov.  1,  1858,  $85. 
What  balance  was  due  Jan.  1,  18^9  ?  Ans.  $241,798. 

13 


194  PERCENTAGE 

Operation. 

$1060  Amount  of  principal  from  Mar.  9,  '55  to  Mar.  9,  '56,  (1  yr.). 

207.74  "         1st  payment  from  Nov.  19, '55  to     "       "  (3fm.). 

852.26       Balance,  forming  2d  principal. 
51.135     Interest  from  Mar.  9,  '56  to  Mar.  9,  '57,  (1  yr.). 

903.395  Amount. 

50.  2d  payment,  being  less  than  interest,  has  no  interest. 

853.395  Balance,  forming  3d  principal. 

64.857  Interest  from  Mar.  9,  '57  to  June  15,  '58,  (1  yr.  3  m.  6  d.). 

918.252  Amount. 

600.  3d  payment,  time  being  more  than  1  year,  has  no  interest, 

318.252  Balance,  forming  3d  principal. 

10.396  Interest  from  June  15,  '58  to  Jan.  1,  '59,  (6  m.  16  d.). 

328.648  Amount  of  3d  principal  to  time  of  last  payment. 

85.85  Amount  of  $85  from  Nov.  1,  '58  to  Jan.  1,  '59,  (2  m.). 

$242,798,  Ans.    Balance  due  Jan.  1,  '59. 

S77.     Annual  Interest. 

III.  Ex.  What  is  the  amount  due  on  a  note  for  $1000,  inter- 
est payable  annually,  if  no  payment  should  be  made  till  the  ex- 
piration of  4  y.  6  m.  12  d.  ? 

The  holder  of  this  note  should  be  allowed  interest  on  the 
interest  from  the  time  it  is  payable  to  the  time  of  settlement, 
in  addition  to  the  interest  upon  the  note. 

The  int.  on  $1000  for  4  y.  6  m.  12  d.  =  $272.00 

"         "     $60  for  3  y.  6  m.  12  d.^ 


2  y.  6  m.  12  d. 


$60   "  6  m.  12  d. 


=  the  int.  on  $60 


ly.  6  m.  12d.  Kor  8  y.  1  m.  18  d 


>  —  29.28 


Principal,     1000.00 


Amount  due,  $1301.28,  Ans. 

The  interest  is  first  taken  upon  the  face  of  the  note  for  the  full 
tinic  •  then  upon  the  $60  due  at  the  end  of  the  first  year  for  the 
balance  of  the  time  for  which  the  note  has  to  run ;  and  so  on  for 
the  other  payments.     Hence  the 

Rule  for  Annual  Interest.  Compute  interest  on  the 
principal  for  the  entire  time  it  is  on  interest ;  compute  interest 


COMPOUND    INTEREST.  195 

also  upon  one  year's  interest  for  the  sum  of  all  the  periods  of 
time  for  which  each  yearly  interest  remains  unpaids  The  sum 
of  the  interests  thus  found  will  he  the  annual  interest. 

Examples. 

1.  What  is  the  annual  interest  of  $200  for  4  y.  6  m.  3  d.? 

Ans.  $59,884. 

2.  What  is  the  annual  interest  of  $334  for  3  y.  8  m.  10  d.  ? 

Ans.  $80,148+. 

3.  What  is  the  annual  interest  of  $118.50  for  5  y.  3  m.  18  d.  ? 

Ans.  $42,588+. 
.  4.  What  is  the  amount  at  annual  interest  of  $175  for  6  y.  2  m. 
25  d.?  ^ws.  $250,821+. 

Note.  — For  New  Hampshire  rule  for  annual  interest  with  partial  pay- 
ments, see  Appendix,  page  334. 


COMPOUND   INTEREST. 

S78,  Compound  Interest  is  interest  on  both  principal  and 
interest,  the  sum  of  the  two  forming  a  new  principal  at  specified 
intervals  of  time. 

Note.  — Interest  may  be  compounded,  or  added  to  the  principal,  annu- 
ally, semi-annually,  or  for  any  period  of  time  agreed  upon. 

379.  III.  Ex.,  I.  What  is  the  compound  interest  of  $212 
for  2  y.  5  mo.  6  d.,  at  6%  ? 


Operation. 
Principal, 
Amount  of  $1  for  1  year, 

$212 
1.06 

«            $212  for  1  year, 

224.72 
1.06 

«            $212    «  2  years, 
"            $1        «   5m.6d., 

238.2032 
1.026 

"            $212    "  2y.  5  m.  6d., 
Principal,  subtracted, 

244.3964832 
212. 

Compound  Interest, 

$32,396+,  Ans. 

196 


PERCENTAGE. 


EuLE  FOR  Calculating  Compound  Interest.  Find  the 
amount  of  the  principal  to  the  time  when  interest  is  first  due ; 
find  the  amount  of  this  sum  for  a  second  period  of  time  as  at 
first,  and  so  on  till  the  entire  periods  of  time  for  which  interest 
is  to  he  compounded  are  exhausted;  find  the  amount  for  the 
balance  of  time  as  in  simple  interest.  This  will  he  the  amount 
at  compound  interest.  To  obtain  the  compound  interest,  subtract 
the  first  principal. 

Examples. 

1.  What  is  the  amount  of  $350  for  3  years,  at  6%  ? 

Ans.  $416,855+. 

2.  What  is  the  compound  interest  of  $250.50  for  4  years,  at 
^%  ?  Ans.  $53,984+. 

3.  What  is  the  amount  of  $1000  for  3  y.  11  mo.,  at  6%  ? 

Ans.  $1256.521+. 

4.  What  is  the  compound  interest  of  $427.56  for  3  y.  7  m. 


6  d.,  at 


Ans.  $100,003. 


5.  What  is  the  compound  interest  of  $350.60  for  2  y.  11  m., 


at  7' 


Ans. 


).558+. 


6.  What  is  the  amount  of  $250  for  1  y.  3  m.  18  d.,  at  5%  per 
annum,  interest  compounding  semi-annually  ? . 

Operation. 
Amount  of  $1  for  6  mo., $1,025 

Multiplied  by 250 

Amount  of  $250  for  1st  6  mo., 256.25 

Multiplied  by  amount  of  $1  for  6  mo.,  ....  1.025 

Amount  of  $250  for  1st  12  mo., 262.656+  . 

Multiplied  by  amount  of  $1  for  3  m.  18  d.,    .         .         .  1.015 

Amount  of  $250  for  1  y.  3  m.  18  d.,        .         .         .      Ans.  $266,595+ 

7.  What  is  the  amount  of  $30  for  1  y.  2  m.,  at  6%  per  an- 
num, interest  compounding  semi-annually?  Ans.  $32,145+. 

8.  What  is  the  compound  interest  of  $800  for  1  y.  1  m.,  at 
6^,  interest  compounding  quarterly?  Ans.  $53,336+. 

9.  What  is  the  compound  interest  of  $240  for  8  m.  15  d.,  at 
10%  per  annum,  interest  payable  semi-annually? 

10.  Find  the  compound  interest  of  $80  from  Sept.  1    1860  to 
Oct.  7,  1861,  at  8  %. 


COMPOUND  INTEREST. 


197 


11.  What  is  the  amount  of  $1400  for  1  y.  2  m.  28  d.,  interest 
compounding  at  the  expiration  of  every  4  months  ? 

12.  Required  the  amount  of  $700  from  Jan.  5,  1864,  to  Nov. 
21,  1865,  at  4%  per  annum,  interest  payable  semi-annually. 

380,     The  process  of  computing  compound  interest  may  be 
shortened  by  the  following 

Table, 

Showing  the  amount  of  $1  or  £1  at  compound  interest  from  1  year 
to  30  years,  at  3,  4,  4^,  5,  6,  and  7^. 


Year. 
1 

3  p.  cent. 
1.030000 

4  p.  cent. 

4|  p.  cent. 

5  p.  cent. 

6  p.  cent. 

7  p.  cent. 

1.040000 

1.045000 

1.050000 

1.060000 

1.070000 

2 

1.060900 

1.081600 

1.092025 

1.102500 

1.123600 

1.144900 

3 

1.092727 

1.124864 

1.141166 

1.157625 

1.191016 

1.225043 

4 

1.125509 

1.169859 

1.192519 

1.215506 

1.262477 

1.310796 

5 

1.159274 

1.216653 

1.246182 

1.276282 

1.338226 

1.402552 

6 

1.194052 

1.265319 

1.302260 

1.340096 

1.418519 

1.500730 

7 

1.229874 

1.315932 

1.360862 

1.407100 

1.503630 

1.605781 

8 

1.266770 

1.368569 

1.422101 

1.477455 

1.593848 

1.718186 

9 

1.304773 

1.423312 

1.486095 

1.551328 

1.689479 

1.838459 

10 

1.343916 

1.480244 

1.552969 

1.628895 

1.790848 

1.967151 

11 

1.384234 

1.539454 

1.622853 

1.710339 

1.898299 

2.104852 

12 

1.425761 

1.601032 

1.695881 

1.795856 

2.012197 

2.252192 

13 

1.468534 

1.665073 

1.772196 

1.885649 

2.132928 

2.409845 

14 

1.512590 

1.731676 

1.851945 

1.979932 

2.260904 

2.578534 

15 

1.557967 

1.800943 

1.935282 

2.078928 

2.396558 

2.759031 

16 

1.604706 

1.872981 

2.022370 

2.182875 

2.540352 

2.952164 

17 

1.652848 

1.947900 

2.113377 

2.292018 

2.692773 

3.158815 

18 

1.702433 

2.025816 

2.208479 

2.406619 

2.854339 

3.379932 

19 

1.753506 

2.106849 

2.307860 

2.526950 

3.025599 

3.616527 

20 

1.806111 

2.191123 

2.411714 

2.653298 

3.207135 

3.869684 

21 

1.860295 

2.278768 

2.520241 

2.785963 

3.399564 

4.140562 

22 

1.916103 

2.369919 

2.633652 

2.925261 

3.603537 

4.430401 

23 

1.973587 

2.464715 

2.752166 

3.071524 

3.819750 

4.740529 

24 

2.032794 

2.563304 

2.876014 

3.225100 

4.048935 

5.072366 

25 

2.093778 

2.665836 

3.005434 

3.386355 

4.291871 

5.427432 

26 

2.156591 

2.772470 

3.140679 

3.555673 

4.549383 

5.807352 

27 

2.221289 

2.883369 

3.282009 

3.733456 

4.822346 

6.213867 

28 

2.287928 

2.998703 

3.429700 

3.920129 

5.111687 

6.648838 

29 

2.356565 

3.118651 

3.584036 

4.116136 

5.418388 

7.114256 

30 

2.427262 

3.243397 

3.745318 

4.321942 

5.743491 

7.612254 

198  PERCENTAGE. 

III.  Ex.,  II.  What  is  the  compound  interest  of  1520,  at  7%, 
for  4  y.  1  m.  24  d.  ? 

Operation. 
Amount  of  $1  at  7^  for  4  y.  by  the  table,        .     .      $1.310796 
Multiplied  by  the  principal, 520 

Amount  of  $520  for  4  yrs., 681.613-f- 

MultipHed  by  am't  of  $1  for  1  m.  24  d.,  .     .     .    .         1.0105 

Amount  of  $520  for  4  y.  1  m.  24  d.,   .     ....       688.769-4- 
Principal  subtracted, 520. 

Compound  interest, .      Ans.  $168,769-}- 

13.  What  is  the  conapound  interest  of  $480  for  7  y.  10  m.  ? 

Ans.  $277.8294-. 

14.  What  is  the  amount  of  $100  for  2  y.  4  m.,  at  7%  ? 

15.  What  is  the  compound  interest  of  $200  for  3  y.  2  m.  6  d.  ? 

16.  What  is  the  amount  of  $221,075  for  3  y.  5  m.,  at  7%  ? 

17.  What  is  the  amount  of  $280  for  1  y.  10  m.  22  d.,  interest 
payable  semi-annually  ? 

18!  What  is  the  amount  of  $50  for  3  y.  13  d.,  at  5%  ? 
19!  What  is  the  compound  interest  of  $896  for  2  y.  6  m.  15  d., 
at  5%  ? 

20!  Find  the  compound  interest  of  $300  for  3  y.  4  m.  12  d.,  at 

21*.  Find  the  amount  of  £58  for  3  y.  5  m. 

Ans.  70  £.  16  s.  1+d. 

22!  Find  the  compound  interest  of  75  £.  9  s.  9  d.  for  4  y.  8  m; 
27  d.,  at  5%. 

23!  What  is  the  difference  between  the  compound  and  simple 
interest  of  $678.25  for  3  y.  6  m.  6  d.  ?  Ans,  $11,488—. 

24!  What  is  the  difference  between  the  compound  and  simple 
interest  of  $100  for  1  y.  4  m.,  the  compound  interest  payable 
semi-annually  ? 

25!  What  is  the  difference  between  the  amount  of  $175.08,  at 
compound  and  at  simple  interest,  from  May  7,  1861  to  Sept.  25, 
1863,  at  7%  ? 


PROBLEMS  IN  INTEREST.  199 

26*  Find  the  difference  between  the  simple  and  compound  in- 
terest of  04  £.  12  s.  6  d.  for  2  y,  6  m.  12  d.,  at  8%  ? 
l^  For  Dictation  Exercises,  sec  Key. 

PROBLEMS   IN   INTEREST. 

381.  Since  interest  is  always  the  product  of  the  three 
factors,  principal^  rate,  and  time^  it  follows  that  to  find  the  time, 
rate,  or  principal,  when  the  interest  and  two  of  the  other  terms 
are  given,  it  is  only  necessary  to  divide  the  interest  by  the  prod- 
uct of  the  two  given  terms. 

383.     To   FIND   THE  Time,  when   the   Interest,   Pkin- 
ciPAL,  AND    Rate   are    Given. 

III.  Ex.     In  what  time  will  f  300  gain  |63  interest  at  6%  ? 

Operatiox.  The  interest  of  $300  for  1  year  at 

Int.  of  ^300  for  1  y.  =$18      e^  j^  $18;  it  will  require  as  many 

1^  )  6^'^  years  for  $300  to  gain  $63  as  $18  is 

T~7  -  contained  times  in  $63,  which  is  3i 

3.0  yrs.  Ans.  .  .       „,         i.,  '        ,  ^ 

times.     Ans.  3|  y.     Hence  the 

Rule.  To  find  the  time,  when  the  interest,  principal,  and  rate 
are  given  :  Divide  the  given  interest  by  the  interest  of  the  principal 
at  the  given  rate  for  1  year. 

Examples. 

What  time  will  be  required 

1.  For  $400  to  gain  $20,  at  6%  ?  -^ns.  10  m. 

2.  For  $500  to  gain  |?60,  at  4%  ?  Ans,  3  y. 

3.  For  168.25  to  gain  1^3.003,  at  6%  ? 

4.  For  #640  to  gain  $07.20,  at  7%  ? 

5.  For  $3000  to  gain  $205,  at  5%  ? 

G-  For  $408  to  gain  $170,  at  7|%  ?  * 

7.  For  1450  to  gain  $192.30,  at  8%? 

8.  For  $280  to  amount  to  $301,  at  5%  ? 
Note.  —  Subtract  $280  from  $301  to  find  the  interest. 

9.  In  what -time  will  $200  amount  to  $400,  at  6%  ? 

10.  In  what  time  will  $500  amount  to  $658.33},  at  6%  ? 


200  PERCENTAGE. 

383*      To   FIND    THE   Kate,  when   the   Intekest,  Time, 
AND    Principal,   are   known. 

III.  Ex.  At  what  rate  per  cent,  will  $250  gain  $25  in  2 
years  ? 

OPERATiojf.  $  If  the  interest  of 

Int.  of  $250  for  2  y.  at  1^,   .     .     $5  )  25  $250  for  2  y.  at   1 

5  per  cent,  is  $5,it  will 

require  as  many  times  1^  to  gain  $25  as  $5  is  contained  times  in  $25, 
which  is  5  times.     Ans.  5^.     Hence  the 

Rule.  To  find  the  rate,  when  the  interest,,  time,  and  principal 
are  given  :  Divide  the  given  interest  by  the  interest  of  the  princi-- 
pal  for  the  given  time  at  1  per  eent^ 

Examples. 
At  what  %  will 

1.  $360  gain  $40.80  in  1  y.  5  m.  ?  Ans.  8^^. 

2.  $100  gain  $33i  in  12  y.  6  m.  ?  Ans.  2f  %, 

3.  $250  gain  $3.75  in  4  m.  ? 

4.  $25  gain  $7.87^  in.  3  y.  6  m.? 

5.  $100  gain  $25in7|y.? 

6.  $48.24  gain  $8J1  in  2  y.  9  m.  10  d.? 

7.  $75  amount  to  $78.75  in  2  y.  6  m.  ? 
Note.  —  ^TS-TS  —  $75  =  $3.75  interest. 

8.  At  what  rate  will  $50  amount  to  $55.25  in  2  y.? 

9.  At  what  rate  wiU  $1000  amount  to  $1058.334  in  10  m.  ? 

384*     To  riNi>   THE   Principal,   when    the  Interest, 
Time,  ani>   Rate   are   known. 

III.  Exi  What  principal  will  yield  $42.50  interest  in  8.  m. 
15  d.  at  6%  ? 

OpEEuVTIOK-.  .f 

Int.  of  $1  for  8  m.  15  d.  at  Q%, .  .  .  $.0425  )  42.50         (1000. 
The  interest  of  $1  for  8  m.  15  d.  at  6^  is  $.0425 ;  it  will  require  aa 
many  dollars  of  principal  to  gain  $42.50  as  $.0425  is  contained  times,  in 
$42.50,  which  is  1000  times.    Ans.  $1000.    Hence  the 


PROBLEMS  IN  INTEREST-  201 

Rule.  To  find  the  j^rincipal,  when  the  interest,  time,  and  rate 
are  known :  Divide  the  given  interest  hj  the  interest  of  1  dollar  at 
the  given  rate  for  the  given  time. 

Examples. 
What  principal  will  gain 

1.  115  in  2  y.  at  6%  ?  Ans.  |125. 

2.  $20  in  4  y.  at  5%  ?  Ans.  $100. 

3.  $76.50  in  2  y.  6  m.  at  3%  ? 

4.  $1,705  in  7  ra.  15  d.  at  4%  ? 

5.  $68,990  in  1  y.  4  m.  24  d.  at  5%  ?. 

6.  $4,128  in  11  m.  14  d.  at  6%  .^ 

Note.  —  (4.128  -^  .057 1).  Reduce  dividend  and  divisor  to  thirds  before 
dividing.     Ans.  $72. 

7.  What  principal  will  be  required  to  gain  $24  in  60  days  at 
2%  a  month? 

8.  What  principal  must  be  on  interest  2  y.  5  m.  29  d.  at  6  %  to 
gain  $89.40  ? 

9.  What  is  the  principal  which  being  on  interest  at  7%  per 
annum,  gains  $62.50  semi-annually.'^ 

S85,   To  FIND  THE  Principal,  when  the  Amount,  Time, 
AND  Rate  are  known. 
III.  Ex.     What  principal  will  amount  to  $17,238  in  2  m.  12 
d.  at  7%  ? 

Operation.  The  amount  of  1  dollar  for  2  m.  12  d.  is 

$17.238 -f- $1,014  =  17.  $1,014;  it  will  require  as  many  dollars  to 
amount  to  $17,238  as  $1,014  is  contained  times  in  $17,238,  which  is 
17  times.     Ans.  $17.     Hence  the 

Rule.  To  find  the  principal,  when  the  amount,  time,  and  rate 
are  known :  Divide  the  given  amount  hy  the  amount  of  1  dollar 
at  the  given  rate  for  the  given  time. 

Examples. 
What  principal  will  amount 

1.  To  $870  in  7  y.  6  m.  at  6%  ?  Ans.  $600. 

2.  To  $537.50  in  2  y.  6  m.  at  6%  ?  Ans.  $467.39/^. 


202  PERCENTAGE. 

8.  To  $2072.25  in  30  da.  at  5%  ?  *  Ans.  $2063.651-|-. 

4.  To  $412  in  90  da.  at  1%  a  month?  A7is.  $400. 

5.  To  $100  in  3  y.  6  m.  at  5i%  ? 

6.  To  $343.75  in  2  y.  1  m.  at  7%  ? 

7.  To  $206.25  in  7  m.  15  da.  at  5%  ? 
I^^  For  Dictation  Exercises,  see  Key. 

PRESENT  WORTH  AND  DISCOUNT. 

386,  This  subject  is  a  practical  application  of  Art.  285.  It 
embraces  all  examples  in  which  it  is  required  to  know  what  sum 
will  equitably  discharge  a  note  or  debt  at  a  given  time  before  it 
is  due. 

SST,  The  Present  Worth  of  any  sum  of  money  due  at  a 
future  time  without  interest,  is  such  a  sum  as  put  at  interest  at 
the  given  rate  w^ill  amount  to  the  debt  when  it  becomes  due. 

It  is  evident  that  where  money  is  worth  6%  a  year,  $106  due 
in  one  year  is  the  same  in  value  as  $100  paid  now  ;  for  $100 
put  at  interest  for  1  year  will  amount  to  $106. 

S88,  Discount  is  that  part  of  an  obligation  which  is  abated 
or  given  up  when  the  payment  is  made  before  it  becomes  due, 
and  should  in  justice  equal  the  interest  upon  the  present  worth 
for  the  given  time. 

389,  III.  Ex.  What  is  the  present  worth  of  $210  due  1 
year  hence,  money  being  worth  5  %  ? 

Here  $210  is  the  amount  of  some  principal  for  1  year  at  5^;  $1^ 
amounts  to  $1.05  in  a  year ;  hence  it  will  require  as  many  dollars  to 
amount  to  $210  as  $1.05  is  contained  times  in  $210,  which  is  200 
times.     Ans.  $200.     Hence  the 

Rule.  To  iind  the  present  worth :  Divide  the  given  sum  by 
the  amount  of  1  dollar  at  the  given  rate  for  the  given  time. 

The  discount  of  the  above  ($210)  is  found  by  subtracting  $200  from 
$210  ;  this  leaves  $10,  which  is  precisely  the  same  as  the  interest  of 
$200  for  1  year  at  5^.     Hence  the 

KuLE.  To  find  the  discount :  Subtract  the  present  worth  from 
the  given  sum. 


PRESENT  WORTH  AND  DISCOUNT.  203 

Examples. 

1.  Find  the  present  worth  of  827,50  for  1  y.  8  m.  at  6%. 
27.50-^1.10  =  25.  Ans.  $25. 

2.  Fmd  the  present  worth  of  $100.90  for  8  mo.  at  6%. 

Ans.  $97,076  -f-. 

3.  Find  the  present  worth  and  discount  of  $200  due  in  3  mo. 
at  G%,  Ajis.  $197.0444-  pres,  worth;  $2.956 —  discount. 

4.  What  is  the  discount  of  $100  for  9  mo.  at  4%  .^ 

A?is,  $2.91 2-f-. 

5.  What  is  the  present  worth  of  $1609.30  for  10  m.  24  d.  at 
5%?  Ans.  $154.0. 

6.  What  is  the  present  worth  of  $175.80  for  9  m.  20  d.  at  6%  ? 

175.80  -^  1.048^  =:  527.40  ^  3.145  =  167.694-f-. 

Ans.  $lG7.G94-(-. 

7.  What  is  the  discount  of  $661,375  for  3  m.  15  d.  at  G%  ? 

Ans.  $11,375. 

8.  What  is  the  present  worth  of  $96,347  for  8  m.  3  d.  at  7%  ? 

A71S.  $92. 

9.  Find  the  present  worth  and  discount  of  $75.50  for  8  m.  10  d. 

Ans.  172.48  pr.  w. ;  $3.02  disc. 

10.  Find  the  present  worth  of  $800.75  for  1  y.  1  m.  10  d. 

Ans.  $750,703+. 

11.  What  is  the  present  worth  of  $75.85  due  in  4  m.  at  5%  ? 

12.  What  is  the  present  worth  of  $221,075  due  in  3  y.  5  m. 
at  7%  ? 

13.  If  a  note  for  $500  be   due  in  2  years  without  interest, 
what  is  its  value  at  the  present  time,  money  being  worth  7%  ? 

14.  What  is  the  present  value  of  $50  due  in  3  y.  13  d.,  inter- 
est being  5%? 

15.  A  note  for  $240  is  dated  June  1,  1860,  due  in  8  m.  15  d. ; 
what  money  will  discharge  it  at  date  ?  Ans.  $230,215-}-. 

16.  A  note  for  $500  is  dated  April  6,  due  in  90  days  ;  what 
money  will  discharge  it  at  date? 

17.  What  would  discharge  the  above  June  23  ?    Ans.  $499+. 

18.  A  note  for  $2000,  dated  July  15,  was  given  for  1  year, 
without  interest ;  what  will  discharge  it  at  date  ? 


204  PERCENTAGE. 

19.  What  will  discharge  it  Oct.  15  of  the  same  year? 

^m.  $1913.875-f. 

20.*  What  sum  paid  down  will  discharge  a, note  of  $500,  due 
in  2^  years,  the  rate  being  5%  ? 

2 it  What  is  the  cash  value  of  a  note  for  ^927.60  on  7  days' 
credit  ? 

22!  What  is  the  value  of  a  note  for  $139.50  Dec.  11,  1863, 
which  is  dated  Sept.  9,  1863,  and  given  for  1  year  ? 


$251.90.  Trenton,  April  1,  1862. 

In  nine  months  from  date,  I  promise  to  pay  J.  Adams,  or 
bearer,  two  hundred  fifty-one  j®^^  dollars,  value  received. 

C.  Quint. 

23!  What  will  discharge  the  above  at  its  date,  the  rate  of  dis- 
count being  6%  ? 

24?  What  will  discharge  the  above  April  16,  1862,  the  rate 
being  7%  ? 

j^^  For  Dictation  Exercises,  see  Key. 

BANK  DISCOUNT. 

390.  Bank  Discount  is  an  allowance  made  to  a  bank  for 
advancing  money  on  a  note  before  it  is  due. 

^91.  Bank  discount  is  the  interest  on  the  face  of  the  not© 
or  its  amount  at  maturity  for  the  time  it  is  discounted  (Art.  272). 

39^,  The  holder  of  a  lote  discounted  at  a  bank  receives  the 
face  of  the  note  minus  the  discount.  This  is  called  the  present 
worth,  the  proceeds,  or  avails  of  the  note. 

III.  Ex.     What  is  the  bank  discount  on  a  note  of  $300  for 
?     What  are  the  avails  ? 

Operatiox. 
The  interest  on  $300  for  4  mo.  is        $6.00 
«  «        «  $300  for  3  d.  is  .15 


"  $300  for  4  m.  3  d.  is  $6.15,  discount. 
$300  —$6.15  =  $293.85,  avails  of  note. 


BANK   DISCOUNT.  205 

Hence  the 

Rule.  To  find  the  Bank  Discount :  Compute  simple  interest 
on  the  given  sum  fourth  e  time  it  is  to  remain  on  interest,  plus  three 
days  of  grace. 

To  find  the  avails  :  Subtract  the  discount  from  the  given  sum. 

Note.  Suppose  the  above  (111.  Ex.)  was  a  6  months  note  dated 
Jan.  5,  which  was  to  be  discounted  at  a  bank  March  5,  the  operation 
would  be  precisely  the  same;  the  note  would  mature f  July  5,  with 
grace,  July  8,  and  would  be  discounted  for  the  time  to  elapse  between 
March  5  and  July  8,  which  is  4  months  and  3  days. 

Examples. 
Find  the  bank  discount 

1.  On  $75  for  30  days.  Ans.  I.412+. 

2.  On  a  90  days  note  for  $500,  dated  May  10,  and  discounted 
June  9.  Ans.  $5.166-|-. 

3.  On  a  60  days  note  for  $256.84,  dated  Oct.  28,  and  dis- 
counted Nov.  12.  Ans.  $2,054-]-. 

4.  On  $1000  for  3  mo.  at  7%.  Ans.  $18,083+. 

5.  What  are  the  avails  of  a  note  of  $700,  discounted  at  a 
bank  for  69  days  ?  Ans.  $691.60. 

6.*  A  trader  buys  900  pairs  of  shoes  at  $.75  a  pair  cash,  and 
immediately  sells  them  at  $.90  on  a  note  payable  in  4  months 
without  interest ;  suppose  he  gets  his  note  discounted  at  a  bank 
for  the  4  months,  what  will  he  have  made  ?  Ans.  $118,395. 

^^  For  Dictation  Exercises,  see  Key. 

39*S«     To    FIND    FOR  WHAT    A  NOTE  MUST  BE   GIVEN,  WHICH^ 
DISCOUNTED    AT    A    BaNK,  WILL    YIELD    A    CERTAIN    SuM. 

III.  Ex.  What  must  be  the  face  of  that  note  which,  being 
discounted  at  a  bank  for  60  days,  will  yield  $148,425? 

The  bank  discount  of  $1  for  63  d.  =r$.0105;  $1 -^  $.0105  p- 
$.9895,  avails  of  $1.     $148,425  ~  $.9895  =  $160,  face  of  note. 

If  $1  were  discounted  at  a  bank,  it  would  yield  $.9895  ;  to  yield 
$148.42,  the  note  must  be  given  for  as  many  dollars  as  $.9895  is  con- 
tained times  in  $148,425,  which  is  150  times.  Ans.  $150, 

t  See  Art.  272,  also  Appendix,  p.  330. 


206  PERCENTAGE. 

Hence  the 

Rule.  To  find  the  face  of  a  note  which,  discounted  at  a  bank^ 
will  yield  a  certain  sum  :  Divide  the  required  sum  hy  $1,  minus 
the  hank  discount  of  %1  for  the  given  time  at  the  given  rate,  and 
the  quotient  will  he  the  face  of  the  note. 

Examples. 

1.  "What  must  be  the  face  of  a  note  that  it  may  yield  $80  when 
discounted  at  a  bank  for  30  days  ?  Ans.  $80.442~(-. 

2.  For  what  must  a  note  on  4  months,  without  interest,  be 
given,  that,  when  discounted  at  a  bank,  it  may  yield  $489.75  ? 

Ans.  S500. 

3.  For  what  must  a  note  be  given,  which  is  to  run  90  days, 
that- it  may  yield  $400? 

4.  What  must  be  the  face  of  a  note  having  60  days  to  run, 
that  it  may  yield  $989.50  ? 

5.  For  what  must  a  note,  dated  Sept.  1,  on  4  months,  be  given 
to  yield  at  its  date  $400,  when  interest  is  7  %  ? 

6.  For  what  must  a  note,  dated  Jan.  1,  payable  in  3  months, 
discount  being  5^%,  be  given,  to  yield  |150?    Ans.  1152.162—. 

7*  For  what  must  a  note  on  6  months  be  written,  to  yield 
$495.85,  when  the  discount  is  7^%  ? 
11^°  For  Dictation  Exercises,  see  Key. 

3d4:*  Miscellaneous  Examples  in  Banking,  &c. 

Note.  — All  examples  in  Present  Worth  or  Discount  should  be  con- 
jsidered  in  True  Present  Worth  or  Discount  (page  202),  unless  Bank  Pres-  ' 
ent  Worth  or  Bank  Discount  is  definitely  stated. 

1.  A  note  for  $500,  dated  July  1,  is  given  for  20  days  without 
interest ;  what  is  its  true  value  July  15  ?  Ans.  $499.50-f-. 

2.  What  will  discharge  the  above,  Aug.  14  ?     (Exact  days.) 

Ans.  $502. 


$200.  Boston,  April  1,  1862. 

Four  months  from  date,  I  promise  to  pay  John  Bills,  or  order 
two  hundred  dollars,  value  received.  John  Orne,  Jr. 


BANK  DISCOUNT.  207 

3.  Suppose  the  above  to  be  a  good  and  true  note,  what  is  it 
really  worth  to  the  holder  in  cash  at  its  date,  money  being  6%  ? 

Ans.  $196,078+. 

4.  What  could  he  get  from  a  bank  ^r  it  at  its  date  ? 

Ans.  $195.90. 

5.  What  would  he  get  for  it  May  1,  by  true  discount?  (Art. 
287.)  Ans.  $197,044+. 

6.  What  ought  he  to  get  for  it  April  1,  1863  ?  Ans.  $208. 

7.  What  is  the  difference  between  the  avails  of  a  note  for 
$200,  payable  without  interest  in  18  months,  whether  it  be  paid 
by  true  or  by  bank  discount?  Ans.  $1,586+. 

8.  What  will  be  the  difference  between  the  true  and  the  bank 
discount  of  a  note  for  $90.50,  due  Feb.  9,  1862,  and  discounted 
June  15,  1861?  Ans.  $.177+. 

$300.  Hartford,  Juhj  15,  1860. 

For  value  received,  I  promise  to  pay  to  the  order  of  myself 
three  hundred  dollars,  in  one  year,  with  interest  after  six 
months.  John  A.  Andrew. 

9.  What  sum  will  the  holder  of  the  above  receive,  if  it  be 
discounted  at  a  bank  Sept.  15,  1860  ? 

10.  What  sum  would  the  holder  of  the  above  receive  at  its 
date  by  true  discount  ? 

11.  What  would  discharge  the  above  note  May  8,  1861  ? 

12.  What  would  be  the  bank  discount  of  the  above  at  its  date? 


$500.  New  Bedford,  Oct.  5,  1860. 

For  value  received,  I  promise  to  pay  Alvin  Dow,  or  order,  five 
hundred  dollars  in  three  months.  Allen  Jones. 

13.  What  cash  must  be  paid  to  discharge  the  above  note  at  its 
date  by  true  present  worth  ? 

14.  What  would  be  the  avails  of  it  at  a  bank  Dec.  5,  1860  ? 

15.  What  would  be  its  real  cash  value  March  17,  1861  ? 

16.  What  would  be  the  true  discount  on  it  Nov.  5,  1860? 

17.  What  would  be  the  bank  discount  of  it  Nov.  5,  1860? 


208  PERCENTAGE. 

18*  What  would  be  due  on  the  above  note  Feb.  15,  1862,  if 
$50  had  been  paid  on  it  at  the  termination  of  each  six  months 
from  its  date,  interest  being  5%  ? 

^9^.     General  Review,  No.  G. 

1.  Reduce  75%,  16|%,  37|-%,  95%,  and  83^%  to  their  low- 
est terms,  and  give  their  sum  in  a  common  fraction. 

2.  If  you  buy  socks  at  14.80  per  dozen  pairs,  and  sell  at  $.50 
per  pair,  what  %  do  you  gain  ? 

3.  For  what  must  apples  which  cost  $1.25  per  bbl.  be  sold  to 
gain  20%  ? 

4.  If  25%  is  lost  by  selling  a  pair  of  boots  at  $4^,  what  was 
the  cost  ? 

5.  What  is  the  simple  interest  of  $300  from  May  5,  1860,  to 
Feb.  2,  1862,  at  1^%,  a  month? 

6.  What  is  the  amount  at  compound  interest  of  $271.36  for 
2y.  6  m.  at  6%? 

7.  What  is  the  present  worth  of  $4508.25  for  11  days,  at  6%? 

8.  What  is  the  bank  discount  of  $450  for  30  days,  at  5%  ? 

9.  What  are  the  avails  of  a  note  of  $100  discounted  at  a  bank 
for  27  days? 

1 0.  What  is  the  amount  at  simple  interest  of  5£.  4s.  6d.  for  2 
years,  at  5  %  ? 

11.  The  interest  of  $400  for  a  certain  time  at  6%  was  $60  ; 
what  was  the  time  ? 

12.  What  principal  at  5%  will  gain  $4.50  in  10  months? 

13.  At  what  Gjo  will  $462  gain  $103.95  in  2  y.  3  m.? 

14.  For  what  must  a  note  be  given,  which,  discounted  at  a 
bank  at  6  %  for  60  days,  will  yield  $1295? 

15.  Given  a  note  for  $2500,  dated  Sept.  5,  1862,  on  which 
were  paid  $50  Jan.  29,  1863,  $500  July  1,  1864.  The  note 
being  on  interest  at  6  %  from  its  date,  what  was  due  Sept.  5, 
1864? 

l^  For  changes,  see  Key. 


COMMISSION,  LIIOKERAGE,  AND  STOCKS.  209 


COMMISSION,  BROKERAGE,  AND    STOCKS. 

396.  Commissioii  is  a  certain  percentage  received  by  a 
commission  merchant  for  transacting  business  as  factor,  or  agent, 
for  another. 

SOT*  Brokerage  is  the  percentage  received  by  a  broker* 
A  Broker  is  one  who  exchanges  money  and  deals  in  stocks  and 
bills  of  credit. 

398.  Stocks  are  Government  Bonds  of  all  kinds,  and  shares 
of  the  capital  invested  in  Banks,  Insurance  Companies,  &c. 

399.  When  stocks  and  money  sell  for  their  original  or  nom- 
inal value,  they  are  said  to  be  at  par  ;  when  they  sell  for  more 
than  their  nominal  value,  they  are  said  to  be  at  an  advance^  above 
par^  or  at  a  premium  ;  when  they  sell  for  less  than  their  nominal 
value,  they  are  said  to  be  at  a  discount^  or  helow  par, 

300.  In  Commission,  the  %  is  estimated  upon  the  sum  ac- 
tually expended ;  in  Brokerage,  upon  the  par  value,  or  an  as- 
sumed value. 

III.  Ex.  My  agent  buys  a  quantity  of  goods  for  $220 ;  what 
is  bis  commission  at  5  %  ? 

1220  X  .05  —  111,  Ans.    Or, 
5  %  :i=  2^0  ;  j(T  of  ^220  =  $11,  Ans, 

Examples. 

1.  What  should  a  commission  merchant  receive  for  selling 
4750  pounds  of  sugar  at  12^  cents  a  pound,  his  commission  being 
1%  ?  "  Ans.  5.937+. 

2.  A  stock  broker  purchases  for  a  person  8  shares  of  stock  in 
a  manufacturing  company  at  $72  a  share ;  what  is  his  commission 
ati%? 

3.  What  is  a  broker's  commission  for  negotiating  a  loan  of 
$4500  at  ^%? 

4.  Dupee  &  Sayles  bought  on  account  of  T.  Winship,  4  sharee 


210  PERCENTAGE. 

of  EjiSex  Company's  stock,  at  $27  each,  their  commission  being 
^%  ;  what  is  Winship's  bill  ?  Ans.  $108.27. 

5.  What  amount  of  current  money  will  be  given  in  exchange 
for  $450  of  that  which  is  at  5%  discount  ?  Ans.  $427.50. 

6.  When  gold  is  at  a  premium  of  25%,  what  must  be  paid  for 
$275  of  gold?  Ans.  $343.75. 

7.  If  an  auctioneer  sells,  on  a  commission  of  8%,  14  chairs  at 
$1.25,  1  bedstead  for  $10,  and  a  miscellaneous  lot  for  $53.79, 
what  sum  will  be  due  the  person  for  whom  he  makes  the  sale, 
his  commission  being  deducted  ? 

8.  What  is  the  commission  on  the  sale  of  200  yards  of  broad- 
cloth at  $4.80  per  yard,  6%  being  paid  for  selling,  and  2^%  for 
guaranteeing  the  sales  ?  Ans.  $81.60. 

Note.  —  The  seller  sometimes  guarantees  the  payment  for  the  goods 
sold ;  for  this  he  is  paid  a  premium. 

9.  AVhat  are  the  net  proceeds  on  the  sale  of  a  lot  of  crockery 
amounting  to  $10650,  commission  being  4^%,  and  lJ-%  being 
allowed  for  guaranteeing  payment?  Ans.  $10011. 

Note.  —  To  obtain  net  proceeds,  deduct  commission. 

10.  What  are  the  net  proceeds  from  the  sale  of  1260  barrels 
flour,  at  $3.50  per  bbl.,  charges  for  freight  and  storage  being  40  c. 
per  bbL,  commission  for  selling  being  2%,  and  for  guaranteeing 
sales  1^%? 

11.  What  are  50  shares  of  railroad  stock  worth,  at  4%  ad- 
vance, the  par  value  being  $100? 

$100  X  1.04  X  50  =r  $5200.  Ans.  85200. 

12.  What  would  be  the  value  of  15  shares  of  the  above  stock, 
at  7  ^c  premium  ? 

13.  What  would  be  the  cost  of  8  shares  of  the  above,  at  a  dis- 
count of  35  %  ? 

14.  What  would  be  the  value  of  4  shares  in  the  stock  of  a  gas 
company,  originally  Mwth  $200,  at  3  %  above  par  ? 

15.  What  would  be  the  value  of  12  shares  of  above  stock,  at 
17%  below  par? 

16.  A  certain  corporation,  wishing  to  increase  their  capital 


COMMISSION  AND  BROKERAGE.  211 

Stock  Without  multiplying  their  shares,  assessed  the  stockholders 
40%  on  the  par  value  of  their  stock,  which  was  $500  per  share  ; 
what  was  assessed  on  a  person  holding  3  shares  ? 

17.  What  was  the  par  value  of  the  stock  per  share  in  the 
corporation  after  the  assessment  was  made  ? 

18.  If  I  buy  10  shares  of  stock,  originally  worth  $100,  at 
18%  above  par,  and  sell  it  at  7%  below  par,  what  do  I  lose  ? 

Ans.  $250. 

19.  What  would  have  been  my  gain  if  I  had  bought  the  above 
at  10%  discount,  and  sold  it  at  a  premium  of  8%  .-* 

20.  Bought  75  shares  in  a  savings-bank,  par  value  being  $50, 
at  61%  advance,  and  sold  at  3^%  above  par;  what  did  I  lose 
on  the  lot  ? 

21.  The  amount  of  the  deposits  in  the  savings-banks  of  Massa- 
chusetts for  1863,  was  $44,785,438.56;  the  ordinary  dividends 
w«re  at  the  rate  of  4|%  of  the  deposits;  what  was  the  total  of 
the  dividends  ? 

301,  To  FIND  THE  Commission  or  Brokerage  and  the 
Sum  invested,  when  both  are  contained  in  a  cer- 
^iN  Sum  sent  to  a  Factor  or  Broker. 

III.  Ex.  I  send  to  my  agent  at  St.  Salvador  $1224;  what 
part  of  this  sum  will  remain  to  invest  in  sugars,  after  deducting 
his  commission  of  2  %  on  what  he  lays  out  ? 

Operation.  Since  the   commission  is 

$1224-^-$1.02  — $1200,  sum  to  invest.  2^  of  the  sum  laid  out, 
$1224— $1200  zzz  $24,  commission.  the  agent  receives  $1.02  for 
every  dollar  which  he  is  to  lay  out.  If  he  receives  $1224,  he  will  have 
as  many  dollars  to  lav  out  as  $1.02  is  contained  times  in  $1224,  which 
is  1200  times.  ^  Ans.  $1200. 

Hence  the  . 

Rule.  To  find  the  sum  invested :  Divide  the  amount  named 
hy  $1  plus  the  commission  on  $1 ;  the  quotient  will  he  the  sum 
invested. 

To  find  the  commission  or  brokerage :  Subtract  the  sum  in^ 
vested  from  the  amount. 


212  PERCENTAGE. 

Examples. 
1    I  have  sent  to  a  commission  merchant  in  New  York  $450, 
of  which  he  is  to  lay  out  what  he  can  in  butter,  after  reserving 
his  commission  of  2  %  on  the  purchase ;  what  is  the  purchase  ? 

Ans.  ^441.176+. 

2.  What  part  of  a  remittance  of  $328.25  will  remain  to  be  in- 
vested after  1  %  of  the  investment  has  been  deducted  ? 

Ans.  $325. 

3.  How  many  barrels  of  flour  at  $5  each  can  a  factor  pur- 
chase with  a  remittance  of  $2575,  after  deducting  his  commission 
of3%?  Ans.  500  hh\. 

4.  How  many  shares  of  stock  at  1 100  each  can  a  broker  pur- 
chase with  a  remittance  of  $520,  allowing  himself  a  brokerage  of 
4%? 

5.  A  real  estate  broker  receives  $2593.75 ;  what  number  of 
acres  of  land  at  $1.25  per  acre  can  he  purchase  with  the  balance 
after  allowing  himself  3|  %  brokerage  on  the  purchase  ? 

6.  Having  remitted  to  my  agent  in  New  Orleans  $891.75,  to 
be  expended  for  sugars,  after  reserving  his  commission  of  2^-%^  I 
received  from  him  29000  pounds  of  sugar ;  what  was  the  cost 
per  pound  ? 

7.  I  have  authorized  a  broker  to  employ  $292.32  in  the  pur- 
chase of  a  certain  stock  for  me,  allowing  him  1^%  commission; 
what  number  of  shares  originally  worth  $100  can  he  purchase, 
if  they  are  now  72  %  below  par  ? 

8*  Wm.  H.  Ladd  sells  for  Chas.  Smith  2500  pounds  of  butter 
at  14  cts.,  and  100  pelts  at  $1.50 ;  from  the  proceeds  he  deducts 
his  commission  of  3%  and  $4  for  cartage,  &c.,  and  with  the  bal- 
ance purchases  for  Smith,  after  deducting  his  commission  of  li% 
on  the  purchase,  a  lot  of  sheeting  at  10  cts.  per  yard;  how  many 
yards  can  he  purchase  ? 

i^^  Tor  Dictation  Exercises,  see  Key. 


INSURANCE.  213 


INSURANCE. 

309.     Insurance  is  security  to  indemnify  for  loss. 

Property  Insurance  indemnifies  for  loss  by  fire,  shipwreck, 
&:c. 

Life  and  Health  Insurance  indemnify  for  loss  of  life  or 
health. 

303,  The  persons  or  company  that  insure  are  called'under- 
writers. 

304:.  A  Policy  is  the  written  contract  between  the  insurer 
and  the  insured. 

30^.  Premium,  is  a  certain  per  cent,  of  the  sum  insured 
paid  to  the  underwriters  for  the  insurance. 

300,     Examples. 

1.  Required  the  premium  for  insuring  a  house  for  $1600  at 
^%.  1%  of  $1600  =  $16;  ^%  of  $1600  =  $8,  A?is. 

2.  What  is  the  insurance  on  $1000  worth  of  furniture  at  ^%, 
including  $1  for  policy  ?  Ans.  $6, 

3.  Insured  |  of  a  store  valued  at  $15000  at  |%  jand  paid  $1 
for  policy.     What  amount  is  paid  ? 

4.  Effected  insurance  on  the  ship  Susan  to  Cadiz  and  back  for 
f  10000  at  2%,  and  on  her  return  cargo,  worth  $7500,  at  1^%  ; 
what  is  the  amount  of  insurance,  including  $1  for  policy? 

Ans.  $313.'b0. 

5.  A  lot  of  clothing  worth  $4000  is  insured  for  §  of  its  value 
at  I  %  ;  if  tlie  goods  are  damaged  by  fire  to  the  amount  of  $500, 
what  will  be  the  net  loss  to  the  underwriters,  making  no  account 
of  interest  ?  Ans.  $480. 

Note.  —  The  underwriters  will  make  good  to  the  insured  his  actual 
loss.     Their  net  loss  will  be  $500  minus  the  premium. 

6.  What  will  be  the  premium  for  insuring  $15500  on  a  school- 
house  for  10  years  at  2|%  ? 

7.  What  would  be  the  loss  to  the  insurance  company  if  the 


214  PERCENTAGE. 

above  building  should  be  destroyed  by  fire,  and  the  insurance  be 
paid  in  6  months  from  the  date  of  the  policy  ?     Ans.  ^15116.84. 

Note.  —  Reckon  interest  on  the  premium  for  6  months. 

8.  Insurance  was  effected  upon  |-  of  a  ship  and  cargo,  valued 
at  85000G,  at  lg%  ;  what  would  be  the  actual  loss  to  the  under- 
writers if  the  ship  and  cargo  should  be  totally  lost  at  sea,  making 
no  allowance  for  interest  ? 

9.  What  would  be  the  actual  loss  to  the  owners  ? 

A?is.  $13203.120. 

307,       To     FIND     FOR    WHAT     SuM     AN    INSURANCE    POLICY 
MUST     BE     TAKEN     OUT,    TO     SECURE    A     CERTAIN     SuM    AND 

COVER  THE  Premium. 

III.  Ex.  For  what  must  a  policy  be  taken  out  to  insure  $500 
on  a  ship's  freight,  and  cover  the  premium  of  2%  .'^ 

Operation. 
1  —  .02  =  .98  Since  the  premium  is  2-%  of  the  policy,  the 

$500 -J- .98  ==$510,204.  property  ($500)  must  be  98^  of  the  policy; 
if  $500  is  98^,  1^  will  be  Jg  of  $500,  and  100^  will  be  100  X  gV 
of  $500,  which  is  $510,204.     Ans.  $510,204.     Hence  the 

Rule.  To  find  for  what  sum  an  insurance  policy  must  be 
taken  out,  to  secure  a  certain  sum  and  cover  the  premium :  Di- 
vide the  sum  to  be  secured  hy  1  minus  the  rate  per  cent,  of  insur- 
ance ;  the  quotient  will  he  the  amount  of  the  policy. 

•  Examples. 

1.  "What  policy  will  cover  $2000  of  property  and  a  premium 
of3%?  Ans  $2061.855+.' 

2.  I  have  loaned  $1140  to  a  friend,  to  be  secured  by  a  policj* 
on  his  life ;  for  what  must  a  policy  be  taken  to  secure  the  sun? 
loaned  and  cover  the  premium  of  5%  also  ?  Ans.  $1200 

3.  Having  adventured  $1800  to  Calcutta,  what  policy  shouh' 
I  take>put  to  secure  both  the  adventure  and  the  premium  of  6%-' 

4.  For  what  must  a  policy  be  taken  out  to  cover  a  loan  of 
$588  and  a  premium  of  121%  upon  it? 

i^"  For  Dictation  Exercises,  see  Key. 


EQUATION  OF  TAYMENTS.  215 


AVERAGE,  OR  EQUATION  OF  PAYMENTS. 

308.  Equation  of  Payments  is  the  process  of  finding  an 
average  time  for  the  equitable  payment  of  several  sums  due  at 
different  times. 

309.  The  Equated  Time  is  the  date  at  which  all  the  items 
may  be  paid  without  loss  to  either  party. 

310.  The  Term  of  Credit  is  the  time  from  the  contract- 
ing of  a  debt  to  the  date  of  its  becoming  due. 

311.    To  FIND  THE  Equated  Time,  when  the  Terms  op 
Credit  begin  at  the  same  Date. 

III.  Ex.  I  owe  P.  Benjamin  two  notes  dated  March  1,  — . 
one  for  $80,  to  be  paid  in  12  months,  the  other  for  $40,  to  be 
paid  in  3  months.  "When,  without  loss  to  either  Benjamin  or 
myself,  can  I  pay  both  notes  at  once  ? 

Interest  Method. 

Operatiox.  I  am  entitled  to  keep  these 

12  months'  interest  on  $80  =:  $4.80     two    notes    till    their    interest 

3         "  "         "      40  i=z       .60      equals  $5.40 ;  if  I  pay  them  both 

Y^Q  g  ^Q      at  once,  it  should  be  at  such  time 

1  of  .1^  of  120  =  .60  ;  .60 )  5!40      ^^^^''  ^^^'''  ^  ^^  '"^'^^  ^^  ^^^^^^^^ 
^        ^°^  for  $120  to  gain  $5.40;  $120 

gains 60  cts.  a  month;  .*.  to  gain 
Mar.  1+9  mo.  z=z  Dec.  1,  Ajis.      *-  ^a    v     mi 

'  ^0.40,  it  will  require  as  many 

months  as  60  cts.  is  contained  times  in  $5.40  =  9  months,  which  added 

to  Mar.  1,  is  Dec.  1.     Hence 

KuLE  I.  To  find  the  equated  time  when  all  the  terms  of 
credit  begin  at  the  same  date:  Find  the  interest  on  each  item 
for  its  time  of  credit ;  divide  the  sum  of  the  interests  hy  the  in^ 
terest  of  the  sum  of  the  items  for  one  month.  The  quotient  wilt 
he  the  equated  time  in  months. 

Add  the  equated  time  to  the  date. 

Note  I.  —  To  obtain  the  interest  for  1  month,  remove  the  desimal  point 
two  places  to  the  left,  and  divide  by  2. 


216  PERCENTAGE. 

Note  II.  —  If  any  item  contains  cents,  reject  tliem  if  they  are  less  than 
50,  and  increase  the  dollars  by  one  if  they  equal  or  exceed  50. 

Product  Method. 

Operation.  The  use  of  $80  for  12  m.  =  the 

80  X  12  =  960  ^^gg  of  |1  fQ^.  ggQ  ^^ .  ^j^g  ^gg  ^^ 

^Q  X     ^  =^120^  $40  for  3  m.  =  the  use  of  $1  for 

120          )       1080  ^^^  ^'     ^^  ^°^'  ^^^  ^-  +  ^^  ^^^ 
120  m.=:$l  for  1080  m.,  M^hich^ 

9  m.  $120  for  ^\-^oi  1080  m.  or  9  m., 

which  added  to  Mar.  1.  zzrDec.  1,  Ans.     Hence 

Rule  II.  Multiply  each  payment  hy  the  number  of  days  or 
months  to  elapse  before  it  becomes  due  ;  divide  the  sum  of  the  prod- 
ucts hy  the  sum  of  the  payments^  and  add  the  quotient  to  the 
date. 

Note.  —  The  examples  in  this  book  are  performed  by  the  Interest 
method. 

Examples. 

—1.  What  is  the  equated  time  for  paying  $50  due  in  5  m.  from 
May  14,  1863,  |35  due  in  4  m.,  and  $25  due  in  2  m.  from  the 
same  date?  Ans.  Sept.  14,  1863. 

^%  B.  Frank  holds  five  notes  against  me,  dated  June  7,  1864 ; 
one  for  |500  on  4  m.,  one  for  $750  on  5  m.,  one  for  $200  on 
12  m.,  one  for  $400  on  2  m.,  and  one  for  $400  on  17  months* 
credit ;  what  is  the  time  at  which  all  should  be  paid  if  paid  in 
one  sum?  Ans.  Jan.  7,  1865. 

-^.  Having  sold  Samuel  Bond  real  estate  to  the  amount  of 
$2000,  lie  gave  me  four  equal  notes  for  it,  dated  Oct.  4,  and  pay- 
able in  5,  6,  9,  and  12  months ;  what  is  the  average  time  for  the 
payment  of  all  the  notes  ?  Ans.  8  m. 

i^A.  What  is  the  average  time  for  paying  $20  due  in  20  days, 
$20  due  in  100  days,  $70  due  in  30  days,  $20  due  in  60  days, 
and  $40  in  70  days  ?  Ans.  1  m.  20  d. 

-6.  April  1,  C.  A.  Brown  purchased  coal  to  the  amount  of 
^5000,  -^  of  which  was  to  be  paid  in  6  months,  ^  in  12  months, 


EQUATION  OF  TAYMENTS.  217 

and  the  remainder  in  9  months  ;  for  what  time  should  a  note 
without  interest,  dated  April  1, 1865,  in  payment  of  all  the  sums, 
be  allowed  to  run,  and  when  should  the  note  be  paid  ? 

Ans,  Jan.  28,  1866. 
~^a  Aowes  B  ^360.25  (Note  2,  ArtSll),^  of  which  is  to  be  paid 
in  7  months,  ^  of  the  remainder  in  8  months,  ^  of  what  then  re- 
mains in  10  months,  and  the  balance  in  4i  months;  in  how  many- 
months  and  days  should  the  whole  be  paid  J'      •    Ans.  6  m.  22  d. 

7.  Sept.  25,  bought  3  parcels  of  goods,  as  follows  :  a  bill 
amounting  to  $225.25  on  12  months'  credit,  a  bill  amounting  to 
$125^.25  on  8  months'  credit,  and  a  bill  amounting  to  $40  on  5 
months'  credit;  what  was  the  mean  time  for  paying  all? 

Ams,  10  m. 

Note,  —  "When  a  sum.  is  paid  immediately,  the  term  of  credit  expires 
instantly,  and  it  will  have  no  corresponding  interest  or  product  in  time. 

8.  A  person  promised  to  pay  $7000,  |-  of  it  immediately,  ^  of 
the  remainder  in  8  months ,  ^  of  what  then  remained  in  22 
months,  and  the  balance  in  16  months  ;  what  is  the  equated  time 
for  paying  the  whole  ?  Ans.  12  m. 
~~~9.  A  merchant  tailor  finds,  on  examining  his  account  with 
Jones  <&  Ca,  May  -5,  that  he  owes  them  for  150  yds.  of  silk,  at 
$.50  per  yd.,  which  is  due  that  day ;  for  2339  yds.  of  cambric,  at 
$.10,  which  will  be  due  in  6  days;  for  12 J-  yds,  broadcloth,  at 
$5.00  per  yd.,  w^hich  will  be  due  in  1 6  days ;  for  50  yds.  of 
doeskin  at  $3,75  per  yd.,  which  will  be  due  in  20  days  ;  wliat  is 
the  average  time  for  paying  the  whole  ?  If  the  tailor  settles  the 
account  by  giving  his  note,  when  should  the  note  begin  to  bear 
interest? 

313.).  To     FIND    THE    E<5UATED    TiME,   WHEN    THE    TeRMS 

OF  Credit  begin  with  different  Dates. 

III.  Ex,    J,  Rives  bought  of  A.  Ainger  the  following  bills 

of  goods : — 

Sept,  15,  a  bill  amounting  to  $100, 

«      30,    «  «  *<  $400, 


218  PERCENTAGE. 

Oct.  8,  a  bill  amounting  to  $250, 
"    10,     "  "         "   $250,    ■ 

What  is  the  equated  time  for  paying  the  whole  ? 

To  equate  the  above  bills,  it  is  necessary  to  assume  a  date 
from  which  to  compute  the  interest  on  the  several  items ;  any 
date  may  be  assumed,  but  the  most  convenient  date  for  examples 
generally,  on  account  of  reckoning  the  time,  will  be  found  to  be 
the  last  day  of  the  month  before  the  earliest  date  at  which  any  item 
becomes  due  ;  this  in  the  above  example  is  Aug.  31. 

OPERATION  BY  INTEREST  METHOD.         g  j      ^^  ^j^e  assumed  ^ate, 

15  days'mterest  on$100:=:    .25        »         oi   -d-  ij  i        ♦.       [ 

Aug.  31,  Hives  would  lose  mterest 

on  the  several  bills  from  Aug.  31 


400=2.00 


^Q    „          ,,          ,,     250=  1662  to  their  respective  dates,  amount- 

■ _l_Jl  ing  in  all   to   $5.50;  .*.  payment 

^^^^      ^-^^  should  be  made  at  such  time  after 

^ofyl^of  1000z=5;  5)        5.50  Aug.  31   as  will   be  required   for 

J^  $1000  to  gain  $5.50,  =  1  m.  3  d., 

1.1  m.  =:  1  m.  3  d.  which  added  to  Aug.  31  is  Oct.  3, 

Aug.  31+1  m.  3  d.  rr  Oct.  3.  -^ns.     Hence  the 

Rule.  To  find  the  equated  time  when  the  terms  of  credit 
begin  with  different  dates :  Assume  that  the  time  for  paying 
all  the  items  is  on  the  last  day  of  the  month  previous  to  the 
earliest  day  at  which  any  item  is  due  ;  find  the  interest  on  each 
item  from  the  assumed  date  to  the  date  when  it  is  due,  and  divide 
the  sum  of  the  interests  by  ^  of-j-hu  (fihe  sum  of  the  items  ;  the 
quotie^it  will  be  the  time  after  the  assumed  date^  in  months,  when 
all  should  be  paid. 

Note  I.  —  .1  month  =  3  d. ;  .03  J  month  =  1  day  nearly,  etc. 

Note  II.  —  Reject  the  fraction  of  a  day  when  it  is  less  than  -| ;  reckon 
it  1  day  when  it  is  ^  or  more. 

"^1.  Find  the  equitable  time  for  the  payment  of  the  following' 
$300,  due  April  1,  1869;  $450,  due  Dec.  1, 1869;  and  $600,  dutf 
March  10,  1870.     (Assumed  date,  March  31,  1869.) 

Ans,  Nov.  25,  186^ 


EQUATION  OF  PAYMENTS. 


219 


^.  Find  the  mean  or  average  time  for  paying  the  follow- 
ing: S12.45,  due  Feb.  10,  1860;  $24.17,  due  Mar.  1,  1860; 
$15,  due  Mar.  14,  1860;  $30,  due  Mar.  16,  1860;  and  $12.70, 
due  Mar.  7,  1860.  Ans.  Mar.  5,  1860. 

Edwin  Foote's  ledger  contains  the  following  account. 


Thomas  Wing, 


Dr. 


Or. 


1861. 

July  1. 

To  Merchandise. 

250 

00 

1862. 

Apr.  1. 

((              i< 

400 

00 

Oct.  2. 

«             {( 

600 

00 

Note.  —  This  account  shows  that  Wing  bought  of  Foote  merchandise 
at  the  times  and  to  the  amount  indicated. 

,.t  3.  Allowing  interest  on  each  item  from  its  date,  what  is  the 

time  from  which  a  note  should  draw  interest  in  payment  of  all  of 

the  above  items  ?  Ans.  May  9, 1862. 

4.  What  is  the  equated  time  for  paying  the  following  bill  ? 

New  York,  Jan.  1,  1861. 
E.  Train  &  Co. 
1860. 

Jan.  20. 
Mar.  15. 
Apr.  12. 


Bought  of  F.  Fogg  &  Co. 
M'd'se  on  3  mo.,  $100. 

«      "  2    "  100. 

"      "  2    "  100. 


The  above  items  will  be  due  as  follows:  April  20,  May  15,  and 
June  12.     Equate  from  these  dates.     Assumed  date.  Mar.  31. 

Ans.  May  16,  1860. 
5.  Equate  the  following :  — 

RoxBURY,  Jan.  1,  1865. 
Mr,  J.  Stow 

1864.  Bo't  of  Z.  Churchill, 

Jan.    5.  M'd'se  on  5  mo.,  $400. 

May  5.         «      "  4    "  600. 

u     16.        «      a  4^    ^  200. 

Ans.  Aug.  8. 


220 


PERCENTAGE. 


6.  If  one  note  should  be  given  for  the  following  three,  when 
should  interest  commence  upon  it  ? 


A  note  for  $200,  dated  May  15,  1864,  on  90  days. 
"     "      "     250,      "     June  1,        "       "   60     « 
«     «      "     700,      «     July  8,         "      «  30     "  rh^i*^'^ 


7.  "What  is  the  mean  time  for  the  payment  of  the  following 
bills  of  goods  purchased  by  Calrow  &  Co.  of  Armstrong  &  Co.  ? 

1856.  June  1,  a  bill  of  $200  on  90  days. 
«         Feb  1,    «       «      800   "  75     " 

«        Apr.  1,    «       "      300   «  60     " 

"         July  1,    "■       "      650   «  40     «  ^P 

1857.  Feb.  1,    "       "    1000   «  20     «,  (W^t     /- 

8!"  What  is  the  equated  time  for  paying  the  following  ? 

Boston,  Juli/  1,  1864. 
J.  P.  Putnam, 

To  Weymouth  Iron  Co.,  Dr. 


-\H(c^ 


liJ 


1S64. 
Jan.  10. 
"  28. 
Feb.  29. 
Mar.  12. 
ApB.     8. 


To  Merchandise  on  3  mo., 
"  "  "   3  mo., 

"  "  "   60  d., 

"  "  "  4  mo., 

«  «  "   90  d., 


$437 
254 
144 
159 


I    300100 


9t  Find  the  equated  time  for  the  payment  of  the  following 
notes  held  by  Page  &  Son  against  AYashington  Manufacturing 
Co. 


A  note  for  $560. 

dated 

Jan.  1,  1856, 

on  5  r 

Qon 

a 

a 

846.15 

a 

Feb.  11,  " 

'*  6 

a 

a 

a 

728.50 

li 

Mar.  20,  " 

«  6 

a 

u 

(( 

400. 

a 

July  30,  « 

"  6 

(( 

*( 

u 

560. 

a 

Sept.  12,  " 

«  8 

a 

n 

u 

600. 

a 

Dec.  18,  « 

«  6 

i( 

« 

n 

500. 

a 

May  10, 1857, 

(I   a 

a 

(( 

« 

350.75 

i( 

«      7,     « 

i(   a 

a 

a 

(( 

820.20 

a 

Apr.  17,    " 

((   a 

a 

^T  For  Dictation  Exercises,  see  Key. 


AVERAGE  OF  ACCOUNTS.  221 

AVERAGE    OF   ACCOUNTS. 

313.     To   Average  an  Account. 

III.  Ex.  I  have  in  my  ledger  an  account  with  F.  E.  Clarke, 
both  the  debt  and  credit  sides  of  which  consist  of  sundry  items. 
The  footing  is  as  follows :  — 

Dr.  side  $250,  averaging  due  Feb.  9, 1863 ;  the  Cr.  side  $300, 
averaging  due  Apr.  4,  1863 ;  at  what  time  should  I  pay  Clarke 
the  balance  ? 

Assuming,  as  in  Art.  312,  Jan.  31,  1863,  as  the  time  for  settling  this 
account,  we  compute  interest  on  each  item  from  this  time  till  it  is  due. 

By  settling  Jan.  31,  1863, 

I  should  lose  63  days'  int.  on        $300  =  $3.15 
Clarke  would  lose  9  days'  int.  on  $250  z=  $.375 
The  balance  due  Clarke  is  $50.    $2,775  int.,  my  net  loss. 

i  of  ^^  of  $50  =  $.25. 

2.775  -^  .25  =  11.1  m.  =  11  m.  3  d. 

Jan.  31,  1863  +  H  m.  3  d.  ;=  Jan.  3,  1864. 

If,  by  settling  at  the  assumed  date,  my  net  loss  of  interest  would  be 
$2.775, 1  shall  be  entitled  to  keep  the  balance,  $50,  till  it  has  gained 
$2,775,  which,  found  by  dividing  it  by  the  interest  of  $50  for  1  month, 
is  11  m.  3  d.;  this  added  to  Jan.  31,  1863,  is  Jan.  3,  1864,  Ans. 

If,  however,  the  dates  were  transposed,  making  Clarke's  $250  due  to  me 
Apr.  4,  and  my  $300  due  to  Clarke  Feb.  9,  by  settling  at  the  assumed  date 
Clarke  would  lose  63  days'  interest  on  $250  ■=  $2.62^, 
I  should  lose  9  days'  interest  on  300  z=      .45 

The  balance  due  Clarke  is  $50.     $2.17^  int.,  C.'s  net  loss. 

If,  by  settling  at  the  assumed  date,  Clarke's  net  loss  of  interest  is 
$2.17^,  he  may  justly  require  me  to  pay  the  balance,  $50,  at  such  time 
prior  to  Jan.  31,  1860,  as  will  be  required  for  $50  to  gain  $2.17^  of 
interest,  which  is  8  m.  21  d. ;  this,  reckoned  back  from  Jan.  31, 1863,  is 
May  10,  1862. 

From  the  above  example  we  deduce  the  following 


222 


PERCENTAGE. 


Rule.  To  equate  an  account :  Assume  that  all  the  items  art 
to  he  paid  on,  the  last  day  of  the  month  previous  to  the  earliest 
day  at  which  any  item  hecomes  due;  Jind  the  interest  on  each 
item  from  the  assumed  date  to  the  date  at  which  it  becomes  due  ; 
find  the  difference  between  the  interest  on  the  Dr.  and  Cr.  sides  of 
the  account;  divide  this  difference  by  the  interest  on  the  balance 
of  the  account  for  1  month  ;  add  the  quotient  to  the  assumed  date 
when  the  larger  side  has  the  more  interest,  and  subtract  it  from 
the  assumed  date  when  the  larger  side  has  the  less  interest. 

314.  Settlement  can  be  effected  earlier  than  the  equated 
time,  by  deducting  interest  from  the  balance  of  the  account  for 
the  time  between  the  equated  time  and  the  desired  time  of  set- 
tlement. It  can  be  effected  later  by  adding  interest.  The  latter 
will  be  necessary  when  the  equated  time  is  already  past.     Or, 

31^.  If  it  be  desired  to  settle  an  account  at  a  specified 
time,  add  interest  to  the  items  due  before  the  specified  time,  and 
subtract  interest  from  those  due  after  the  specified  time  ;  the  dif- 
ference between  the  total  of  the  Dr.  and  Cr.  items  plus  or  minus 
their  interest,  will  be  the  balance  due. 


310*     Examples. 
1.  When  can  the  balance  of  the  following  ledger  account  be 
paid  without  loss  to  either  party  ? 


Dr. 


Edward  C.  Damon. 


Cr. 


1863. 
Apr.    1 
July    8 

To  Cash,          $1000 
^'  Mdse.,            118 

00 

98 

1863. 
Apr.  14 
Aug.  10 

By  Mdse., 
"  Real  Estate, 

$1392  59 
94  33 

This 

mon  re( 

Apr. 

July 

account  shows  that 
leives, 
1,  $1000. 
8,       118.98. 

Da- 

And 

April 
Aug. 

that  he  is  credi 
14,  with  $139 
10,      «           9 

ted, 

2.59. 

4.33. 

We  assume  March  31  for  settling  the  account. 


AVERAGE  OF  ACCOUNTS. 


223 


Data. 
Apr.  1, 
July  8, 

Operation. 
Br.                                                 Cr. 

mem.      Days.           Int.              Date.         Item.       Days. 
$1000          1     $  .166f       Apr.  14,  $1393         14 
119       99        1.9635        Aug.  10,        94       132 

Int. 
$3.25 
2.068 

1119                  2.130+ 

1487 
1119 

5.318 
2.130 

Balance  of  %,  $368    Bal.  of  int. 
^^  of  $368  =  1.84 ;  3.188  +-  1.84  =  1.73  =  1  m.  22  d 
Mar.  31  +  1  m.  22  d.  =z  Ans.  May  22,  1863. 

$3,188 

2.  Add  3  months'  credit  to  each  item  in  the  following,  and 

equate  the  %. 

Dr.                           a.  B.  in  %  with  C.  D. 

Ck. 

1863. 
Jan.    1 
Mar.31 
May  30 

To  Bal  Ledger  B., 
"  Heal  Estate, 
"     do.       do. 

$50 
50 
55 

00 
00 
00 

1863. 
Mar.    3 

May  27 

By  Mdse., 

$50 
50 

00 
00 

Assumed  date,  Mar.  31,  1863. 


Ans.  May  12,  1863. 


^.  When  is  the  balance  of  the  following  ^  due .'' 
Dr.  Smith,  Dove,  &  Co- 


Cr. 


1862. 
Jan.  6 
Feb.  7 


1862. 

[dse.,30d.er., 

$600 

00 

Jan.    1 

do.    60d.  cr., 

840 

Mar.  16 

By  R.  Estate,  90d.cr., 

"  Cash, 


$500 
300 


Assumed  date  Jan.  31,  1862.  Atis,  Feb.  25,  1862. 

4r  When  is  the  balance  of  the  following  account  due  ? 


Db- 


Day,  Wilcox,  &  Co. 


Cr. 


1864. 

July  21 

To  Mdse. 

90  da.. 

Aug.  15 

"      do. 

60  da., 

Aug.  31 

''      do. 

4  mo.. 

Oct.  17 

''   Cash, 

$173 
13 

81 


230  00 


1864. 
June  25 
June  30 
Aug.  20 
Sept.  12 


By  Mdse. 

,  30  da.. 

$500 

00 

"      do. 

60  da., 

52 

71 

"      do. 

4  mo., 

16 

48 

"      do. 

30  da., 

102 

10 

A71S.    Dec.  26,  1863. 
317,    To  find  the  equitable  time  for  the  payment  of  the  bal- 
ance of  a  debt,  when  partial  payments  are  made  before  the  del/' 


224 


PERCENTAGE. 


is  due :  Make  the  whde  debt  the  Dr.  side  of  an  account,  and  the 
partial  payments  the  Or.  side. 

5.  A  holds  a  note  against  B,  dated  Nov.  14,  1864,  for  $620, 
due  7  months  hence,  without  interest.  On  this  note  B  paid  A 
$220  Jan.  14, 1865,  and  $100  Feb.  14, 1865 ;  what  is  the  equated 
time  for  paying  the  balance?  Ans.  Nov.  16,  1865. 

6.  T.  Ropes  owes  R.  Treat  $250,  due  May  29.  If  he  should 
pay  $50  on  the  29th  of  April  previous,  when  should  he  pay  the 
balance  ?  Aiis.  June  7. 

7.  A  fanner  purchased,  on  the  1st  day  of  April,  1864,  3  acres 
of  land  at  $183  per  acre,  agreeing  to  pay  for  it  in  7  months;  if 
he  should  pay  $50.75  at  the  date  of  the  purchase,  $148.25  in  4 
months,  and  $150  in  3  months,  in  what  time  should  he  pay  the- 
balance  ?  i-         j  u 

8.  A  owes  B  $2000  Oct.  5  ;  if  he  should  pay  $1200  of  it  Sept. 
8,  at  what  time  should  the  balance  be  paid  ? 

9f  J.  Edwards  owes  J.  Adams  $1200  on  a  note  dated  Oct.  9, 

1863,  payable  in  4  months  without  interest ;  if  Edwards  should 
pay  Adams  $250  on  this  note  Jan.  16,  1864,  and  |400  Feb.  9, 

1864,  when  should  the  balance  of  the  note  be  paid? 

10.  Bought  a  lot  of  land  for  $800,  for  which  I  gave  my  note, 
dated  May  7, 1864,  payable  in  6  months ;  June  28,  I  paid  $158 ; 
Aug.  7, 1  paid  $320.60,  and  Sept.  7,  $179.40;  when  should  the 
balance  be  paid  ? 

lit  Find  the  equated  time  for  the  settlement  of  the  following 
account :  — 

Robertson  &  Reynolds  in  %  with  James  Loring  &  Co. 

Dr.  Cit. 


~1864. 

18^. 

July  12. 

To  Balance, 

$562 

IT 

July  18. 

By  Cash, 

$480 

00 

"     20. 

"   Mdse.  on  4  mo. 

1524 

82 

"    27. 

"  Note  on  90  d. 

1218 

65 

Aug.   8. 

"       "       "  2  mo. 

210 

00 

Aug.  20. 

"  Real  Estate, 

600 

00 

Sep.  30. 

"       "        "  4  mo. 

783 

25 

Sep.  30. 

"  Cash, 

459 

50 

Nov.  25. 

"  Bill  due, 

286  58 

Oct.  28. 

"Draft  at  eOd.f 

425 

00 

Dec.    1. 

((     it      (( 

424 

60 1 

iDec.    1. 

"  Cash, 

185 

20 

^p*  For  Dictation  Exercises^  see  Key. 

t  Allow  S  days  of  grace. 


TAXES.  225 


TAXES. 


A  Tax  13  a  sum  of  money  assessed  upon  a  person  or 
upon  property  for  public  purposes. 

319.  A  Poll  Tax  is  a  sum  assessed  upon  each  male  citizen 
liable  to  be  taxed,  without  regard  to  his  property.  The  persons 
thus  taxed  are  called  the  polls. 

3^0«  Real  Estate  consists  of  immovable  property,  such  as 
houses,  lands,  &;c. 

SSI*  Personal  Property  consists  of  movable  property, 
such  as  money,  stocks,  cattle,  ships,  &c. 

3SS.  Assessors  are  officers  appointed  to  levy  taxes.  It  is 
their  duty  to  ascertain  the  value  of  the  taxable  property  and  the 
number  of  polls,  and  to  apportion  the  tax  to  each  person. 

3S3.  III.  Ex.  The  whole  state,  county,  and  town  tax  of 
Oxford  for  the  year  1862,  was  $5300  ;  the  value  of  the  real  es- 
tate and  personal  property  is  $1250000;  there  are  200  polls  in 
the  town,  each  of  which  is  taxed  $1.50.  What  is  the  tax  on  $1, 
and  what  is  J.  Swan's  tax,  who  has  $3000  of  real  estate  anc^ 
$1000  of  personal  property,  and  who  pays  1  poll  tax  ? 

Opkration. 

$1.50  X    200  =  $300,  amount  of  poll  taxes. 
$5300  —  $300  :=  $5000,  property  tax. 
$5000  -^-  $1250000  =^  4  mills,  tax  on  $1  of  property. 
$3000  -f  $1000  ZI3  $4000,  Swan's  taxable  property. 
$4000  X  $-004  —  $16.00,  Swan's  property  tax. 
$16.00  +  $1.50  z=z  $17.50,  Swan's  whole  tax. 
Hence  the 

Rule  for  Apportioning  Taxes,  Multiply  the  tax  on  one 
poll  hy  the  number  of  polls,  and  subtract  the  product  from  the 
whole  tax ;  divide  the  balance  by  the  taxable  property  i  the  quO' 
tient  is  the  tax  on  $1.  Multiply  each  person's  taxable  property 
by  the  tax  on  $1,  and  add  his  poll  (ax^  or  taxes^  if  he  have 
any. 


226  PERCENTAGE. 

Examples. 

1.  The  whole  tax  of  the  town  of  H.  is  $70352  ;  the  valu- 
ation of  the  town  being  $9329000,  the  number  of  polls  being 
3366,  each  taxed  $1.50,  what  is  the  tax  upon  $1  and  what  is 
the  tax  on  the  following  named  tax-payers  ? 

A  has  property  amounting  to  $8500,  and  pays  1  poll. 
B  has  property  amounting  to  $3570,  and  pays  1  poll. 
C  has  property  amounting  to  $5800,  and  pays  0  polls. 
D  has  property  amounting  to  $1000,  and  pays  2  polls. 
E  has  property  amounting  to  $2800,  and  pays  3  polls. 

Ans.  7  mills  on  $1 ;  A's  tax,  $61 ;  B's  tax,  $26.49. 

2.  The  town  of  L.  votes  to  raise  a  tax  of  $8343.20 ;  the  val- 
uation of  the  town  is  $2000000 ;  there  are  1679  polls,  each  taxed 
$.80 ;  what  is  the  tax  on  a  dollar,  and  what  is  the  tax  of  J.  L. 
Partridge,  who  has  $1500  of  real  estate  and  $3000  of  personal 
property,  and  pays  two  poll  taxes  ? 

A71S.  3i  mills  on  $1 ;  Partridge's  tax,  $17.35. 

3.  What  would  be  the  tax  upon  a  non-resident  who  had  prop- 
erty in  the  above-named  town  of  L.  to  the  amount  of  $15225  .-^ 

Ans.  $53,287+. 

4.  The  state  tax  of  a  certain  town  is  $3093 ;  the  county  tax 
is  $5110;  the  town  tax,  $33860;  the  valuation  of  the  town  is 
$6700000 ;  there  are  2542  polls,  each  taxed  $1.50.  What  is 
the  tax  on  $1,  and  what  is  the  tax  on  a  person  having  $4500  in 
real  estate,  $2750  in  personal  property,  and  who  pays  one  poll 
tax?  Ans.  $.005^^4  on  $1 ;  $42.889ff. 

>6.  There  is  a  town  whose  valuation  is  $1100000,  in  which 
there  are  300  polls,  each  taxed  $1.20 ;  the  tax  to  be  raised  is 
$9600.  What  is  the  tax  on  $1,  and  what  is  the  tax  of  a  person 
having  $4000  in  real  estate,  an  annual  income  of  $3000,  on  all 
above  $800  of  which  he  is  taxed  as  for  personal  property,  and 
who  pays  three  poll  taxes  ?  Ans.  8|  m.  on  $1 ;  $55.68. 

6.  The  valuation  of  a  certain  town  in  real  estate  is  $3200000 ; 
in  personal  property  $1186000 ;  the  tax  to  be  raised  is  $31579.20, 


TAXES. 


227 


one  sixth  of  which  is  to  be  levied  upon  the  polls,  of  which  there 
are  3096.     AVhat  is  the  tax  on  $1,  and  what  on  each  poll? 

Ans,  G  m.  on  $1 ;  poll  tax,  $1.70. 

Assessors  commonly  construct  a  table  showing  the  tax  on  $1, 
72,  $3,  &c.;  from  which  they  compute  the  individual  taxes. 

7.  The  valuation  of  a  certain  town  is  $11522400;  the  tax  to 
be  raised  is  $108391.GO ;  there  are  3350  polls,  each  taxed  $1.40. 
Find  the  tax  on  $1,  and  perform  the  remaining  examples  by  the 
following 

Table, 

Showing  the  tax  on  various  sums  at  the  rate  of  $.009  on  $1. 


$1  pays 

$.009 

$10 

pays 

$.09 

$  100 

pays 

$.90 

2   " 

.018 

20 

u 

.18 

200 

u 

1.80 

3   " 

.027 

30 

a 

.27 

300 

« 

2.70 

4   « 

.036 

40 

« 

.36 

400 

u 

3.60 

5   « 

.045 

50 

« 

.45 

500 

« 

4.50 

6   " 

.054 

60 

it 

.54 

600 

(( 

5.40 

7   « 

.063 

70 

(( 

.63 

700 

(t 

6.30 

8   « 

.072 

80 

(( 

.72 

800 

u 

7.20 

9   « 

.081 

90 

a 

.81 

900 

(t 

8.10 

10   " 

.09 

100 

« 

.90 

1000 

t( 

9.00 

8.  At  the  above  rate,  what  is  A's  tax,  he  being  assessed  for 
$4250,  and  paying  2  polls  ? 

Operation. 
$4000  pays  $36. 

200     '♦         1.80 

50     "  .45 

2  polls     "        2.80 


Ans.  $41.05,  A's  tax. 

Find  the  tax  of  the  following  at  the  above  rate  :  — t 
9.  Of  B,  who  is  assessed  for  $2800  and  1  poll. 

10.  Of  C,  who  is  assessed  for  $7850  and  3  polls. 

11.  Of  D,  who  is  assessed  for  $1565  and  1  poll. 

12.  Of  E,  who  is  assessed  for  $906^  and  2  polls. 


228  PERCENTAGE. 

13.  Of  F,  who  is  assessed  for  $6555  and  1  poll. 

14.  Of  G,  who  is  assessed  for  $5687  and  1  poll. 

15.  Of  H,  who  is  assessed  for  $10793  and  3  polls. 

16.  Of  I,  who  is  assessed  for  $3384  and  1  poll. 

17.  Of  J,  who  is  assessed  for  $4597  and  1  poll.       ' 

18.  Of  K,  who  is  assessed  for  $8979  and  2  polls. 

19.  Of  L,  who  is  assessed  for  $2972  and  1  poll. 

20.  Of  M,  who  is  assessed  for  $1000  and  1  poll 

21.  Of  N,  who  is  assessed  for  $6587  and  2  polls. 

22.  Of  O,  who  is  assessed  for  $7572  and  2  polls. 

23.  Of  P,  who  is  assessed  for  $2956  and  1  poll. 
^^  For  Dictation  Exercises,  see  Key. 

CUSTOM  HOUSE  BUSINESS. 

334:*  Custom  Houses  are  places  where  Government  Offi- 
cers collect  duties. 

3S^.  Duties  are  taxes  upon  imports  and  upon  the  tonnage 
or  weight  which  a  vessel  may  carry.  They  are  of  two  kinds, 
specific  and  ad  valorem.  They  furnish  a  revenue  for  the  gov- 
ernment. 

3S6.  An  Invoice  is  a  list  of  imported  goods,  showing  their 
quantity  and  price. 

3^7 •  A  speeiflo  duty  is  a  tax  upon  each  article  of  a  cer- 
tain kind,  without  regard  to  its  value. 

328.  An  ad  valorem  duty  is  a  certain  per  cent,  of  the  cost' 
of  goods,  estimated  upon  the  invoice. 

3^9*  Leakage  and  Breakage  are  allowances  for  loss  from 
the  leaking  and  breaking  of  bottles,  boxes,  (Sec. 

330*     Tare  is  an  allowance  for  the  weight  of  boxes,  &;c, 

331.  Gross  Weight  is  the  weight  of  goods  including  what- 
ever is  used  for  packing. 

33^,     Net  Weight  is  the  weight  of  the  goo<^s  alone. 


CUSTOM  HOUSE  BUSINESS.  229 

333.     Examples. 

1.  What  is  the  net  weight  of  120  boxes  of  raisins,  gross 
weight  being  31  lbs.  each,  the  tare  being  6^  lbs.  per  box  ? 

Ans.  2940  lbs. 
Operation. 
31  —    6^  ==  24|,  net  weight  of      1  box. 
24^  X  120  —  2940,  «       "       "    120  boxes. 

2.  What  is  the  duty,  at  5  cents  per  lb.,  on  the  net  weight  of  the 
above  ?  Ans.  $147. 

3.  What  is  the  specific  duty,  at  15  cents  per  gallon,  on  25  bar- 
rels spirits  turpentine,  containing  32  gallons  each,  5^  being 
deducted  for  leakage  ?  Ans.  $114. 

4.  What  is  the  duty,  at  25  %,  on  75  boxes  of  tin,  112  lbs. 
per  box,  invoiced  at  7  cents  per  pound,  tare  being  6  lbs.  per  box? 

Ans.  $139,125. 

5.  What  is  the  duty  on  100  dozen  watch  crystals,  at  35  5^,  in- 
voiced at  $1  per  dozen,  3%  being  allowed  for  breakage? 

Ans.  $33.95. 

^     6.  What  is  the  duty,  at  20  %  ad  valorem,  on  an  invoice  of  24 

boxes  of  tea,  gross  weight  being  1284  lbs.,  8  lbs.  for  tare  being 

allowed  on   each  box,  the  cost  of  the  tea  being  38  cents  per 

pound?  Ans.  $82,992. 

7.  I  have  imported  3  tons,  3  cwt.  3  qrs.  7  lbs.  of  steel  invoiced 
at  20  cents  per  pound ;  8  %  being  allowed  for  damage,  what  is 
the.  duty  at  20%?  t 

8r  What  is  the  cost  at  the  store  of  5  hhds.  of  sugar,  weighing 
gross  255 G  lbs.,  which  was  bought  in  Havana  for  $178.92,  and 
on  which  is  paid  $35.75  for  freight  and  carting,  and  2^  cents  per 
pound  for  duty  after  deducting  15  %  for  tare  ? 
h~ — ^  d*  Find  the  cost  of  two  cases  of  gum  arable,  at  21£.  5s.  per 
cwt.,  duty  30  %  ad  valorem ;  the  gross  weight  of  the  first  being 
1  cwt.  3  qrs.  20  lbs.,  of  the  second  1  cwt.  1  qr.  10  lbs.,  35  lbs." 
being  allowed  for  the  weight  of  each  case.        Ans.  7d£  Os.  2fd. 

I^^  For  Dictation  Exercises,  see  Key. 

*  See  Art.  1G4,  Note. 


230  PERCENTAGE- 


DRAFTS  AND  BILLS    OF  EXCHANGE. 

334.     A  Draft  is  a  written  order,  directing  one  person  U 
pay  money  to  another.     The  following  is  a  simple  form  of  a 

Draft. 


$100.  Baltimore,  April  4,  18G4. 

Thirty  days  after  sight,  pay  to  Samuel  Price,  or  order,  One 
Hundred  Dollars,  and  charge  the  same  to  my  account. 

Charles  Smith. 
To. Brewer  &  Tileston, 
Publishers,  Boston. 

33^.     The  Drawer  is  the  person  who  signs  the  draft. 

336.  The  Drawee  is  the  person  to  whom  the  draft  is  -iid' 
dressed. 

33 T.  The  Payee  is  the  person  in  whose  favor  the  draft  5:* 
drawn. 

Note.  —  In  the  above,  Charles  Smith  is  the  Drawer,  Brewer  &  Tileston 
are  the  Drawees,  and  Samuel  Price  is  the  Payee. 

338.  The  Holder  of  a  draft  is  the  person  who  has  legal 
possession  of  it. 

339.  The  Endorsement  of  a  draft  is  the  writing  upon  its 
back,  by  which  the  payee  transfers  his  right  in  it  to  another 
person. 

34:0.  If  the  drawee  does  not  pay  the  money  when  the  draft 
is  presented,  he  may  signify  his  acceptance  of  it  by  writing  his 
name  on  its  face,  after  the  word  "  Accepted,"  by  which  act  he 
becomes  responsible  for  its  payment. 

Bills  of  Exchange. 

341.  A  Bill  of  Exchange  is  a  draft  used  by  merchants 
for  the  discharge  of  debts  payable  at  a  distance. 


DRAFTS  AND  BILLS   OF  EXCHANGE.  231 

34^*  Bills  of  Exchange  are  Inland,  or  Domestaic,  when 
they  are  drawn  and  payable  in  the  same  country. 

34LS.  Bills  of  Exchange  are  Foreign  when  they  are  drawn 
in  one  country  and  payable  in  another. 

Illustration.  —  Suppose  A,  in  New  York,  ships  butter  to  B,  in  Liv- 
erpool, to  the  amount  of  $1000.  He  makes  a  draft  on  B  to  pay  to  him- 
self, or  "  bearer,"  or  "  order,"  the  $1000  due.  But  C,  in  Boston,  wishes 
cutlery  from  D,  in  Sheffield,  to  the  same  amount.  So  he  buys  A's  draft, 
paying  its  value  in  United  States  money,  and  sends  it  to  D.  D  receives 
it  and  presents  it  to  B,  in  Liverpool.  B,  having  received  his  butter  in 
good  condition,  accepts  the  draft,  and  when  the  time  comes  for  pay- 
ment of  the  money,  pays  it  to  D,  of  Sheffield,  in  English  currency. 
Thus  A  receives  payment  for  his  butter,  and  D  for  his  cutlery,  without 
the  risk  or  inconvenience  of  sending  the  money  from  one  country  to 
another. 

34:4:.  Bills  of  Exchange  thus  bought  and  sold  are  ^aid  tu 
be  negotiable,  or  marketable. 

•S4:«i»  When  the  value  of  the  goods  sent  from  the  United 
States  to  another  country —  England  for  example — is  greater  than 
the  value  of  those  received  from  England,  more  money  is  due  to 
us  from  the  English  merchants  than  is  due  to  them  from  our 
merchants,  and  we  hold  more  bills  against  England  than  are 
needed  to  pay  our  debts  ;  consequently,  bills  become  cheap,  and 
exchange  is  at  a  discount.  When  the  value  of  the  goods  im- 
ported from  England  exceeds  the  value  of  those  sent  to  England, 
our  merchants  hold  fewer  bills  against  England  than  are  needed 
to  pay  their  debts,  and  bills  thus  become  dear,  and  exchange  is 
at  a  premium.  The  current  price  of  exchange  is  called  the 
Course  of  Exchange. 

34:6,  A  Set  of  Exchange  consists  of  two  or  more  drafls 
of  the  same  tenor  and  date,  the  payment  of  either  one  of  which 
cancels  the  other  one  or  two. 

To  provide  against  accident  in  the  transmission  of  a  drafl,  i\  ia 
customary  to  send  two,  at  least,  of  a  set  by  different  model  ©f 
conveyance,  or  at  different  times. 


232  PERCENTAGE. 

INLAND   OR  DOMESTIC  EXCHANGE. 

34L7*     To  FIND  THE  Value  of  an  Inland  Draft. 

III.  Ex.,  I.  "What  must  be  paid  in  St.  Louis  for  a  draft  on 
Philadelphia,  for  $2500,  payable  at  sight,  exchange  being  1^% 
premium  in  favor  of  Philadelphia  ? 

Operation.  If  exchange   is   11^  premium, 

$2500  X  1.015  =  $2537.50,  Ans.    the  exchange  value  of  $1  is  $1,015 
and  the  price  of  $2500  will  be  $2500  X  1.015  r=  $2537.50,  Ans. 

III.  Ex.,  II.  What  must  be  paid  in  New  Orleans  for  a  draft 
on  New  York  for  $1500,  payable  in  60  days  after  sight,  exchange 
being  1%  premium  ? 

Operation. 
$1500  ^1.01      =:  $1515.,  Exchange  value  of  $1500  at  sight. 
$1500  X     .0105  =        15.75,  bank  discount  of  $1500  for  63  days. 
Alls.  $1499.25,  cost  of  draft. 
Hence  the 

Rule.  To  find  the  value  of  an  inland  draft :  Multiply  the 
face  of  the  draft  hy  the  exchange  value  of  %1.  If  the  draft  is 
payable  after  sight.,  deduct  from  the  product  the  hank  discount  of 
the  face  of  the  draft  for  the  given  time  and  grace. 

Examples. 

1.  What  is  the  value  of  a  draft  on  Boston  for  $2500,  when  ex- 
Change  is  at  a  premium  of  |  %  ?  Ans.  $2506.25. 

2.  What  must  I  pay  for  a  draft  on  New  York  for  $700  at 
12  days  without  grace,  exchange  being  at  |%  premium? 

Ans.  $699,475. 

3.  Bought  a  bill  on  New  Orleans  for  $400,  at  a  discount  of 
1  %  ;  what  did  I  pay  ?  Ans.  $398. 
*^4.  Messrs.  B.  &  T.,  of  Boston,  sold  a  draft  for  $206.59  on 
Billings  &  Son,  of  Baltimore,  at  30  days'  sight,  discount  being 
^%  ;  what  did  they  receive  for  it? 


EXCHANGE.  233 

S4:8#     To  FIND  THE  Face  op  a  Draft. 

III.  Ex.  What  is  the  face  of  a  30  days'  draft  on  Cincinnati 
at  1  %  discount,  which  can  be  purchased  at  New  York  for  $200  ? 

Operatiox. 
$1  — $.01  =1  $.99,      exchange  value  of  $1  at  sight. 
.0055,  bank  discount  of  $1  for  33  days. 
.9845,  exchange  value  of  $1  at  30  days. 
$200  -^  .9845  =:  $203,148+,  Ans. 

The  exchange  value  of  $1  of  the  draft  will  be  $.9845 ;  if  $1  can  ho 
purchased  for  $.9845,  as  many  dollars  can  be  purchased  for  $200  as 
$.9845  is  contained  times  in  $200,  which  is  203.148-}-  times.  Ans. 
$203,148-1-.     Hence  the 

Rule.  To  find  the  face  of  a  draft  which  may  be  purchased 
for  a  given  sum :   Divide  the  given  sum  ly  the  exchange  value 

of  n. 

Examples. 

1.  Wishing  to  remit  to  my  correspondent  at  St.  Louis  the  net 
proceeds  of  a  lot  of  wheat,  amounting  to  $1275,  I  purchase  with 
that  sum  a  draft  at  1  g  %  discount ;  required  the  face  of  the  draft. 

Ans.  $1289.506+. 
—  2.  What  is  the  face  of  a  draft  for  15  days,  which  may  be  pur- 
chased for  $1050,  at  1^%  premium?  Ans.  $1037.549+. 
-^3.  What  is  the  face  of  a  bill  on  Boston  for  60  days,  at  -^  % 
premium,  which  may  be  bought  for  $3000  ? 


y 


FOREIGN  EXCHANGE. 


34:9*  The  method  of  computing  foreign  exchange  is  similar 
to  that  of  computing  inland  exchange,  except  that  the  currency 
of  one  country  must  be  reduced  to  that  of  another. 

3cS0*  The  value  of  1£  sterling,  which  is  the  English  sov- 
ereign, compared  with  the  old  United  States  coin,  equals  $4.44|. 
But  Congress  has  from  time  to  time  reduced  the  weight  and  purity 
of  United  States  coins,  making  their  value  as  metals  less  than  their 
value  as  coins,  that  they  might  not  be  used  for  transportation  or 


234  PERCENTAGE. 

the  arts,  and  has  established  the  legal  value  of  the  pound  sterling 
at  $4.84.  The  intrinsic  value  of  the  pound  is  $4,861.  The  com- 
mercial value  varies  from  $4.83  to  $4.87,  as  it  is  in  greater  or 
less  demand. 

S5t,  Exchange,  however,  is  reckoned  upon  the  old  or  nom- 
inal value  of  the  pound  ($4.44|),  and  the  present  value  is  said 
to  be  at  about  9  %  premium.  Thus,  when  exchange  on  England 
is  quoted  at  10  or  11%  premium,  it  is  really  only  at  about  1  or 
25^  premium  upon  the  real  value. 

309.    Examples   in  Heduction  of  Currency  and  Ex- 
change. 

1.  What  is  the  nominal  value  of  £250  sterling  expressed  in 
United  States  money? 

$4.44|  =  rV.     ^^/-4^  =  $1111.11^,  Ans. 

2.  What  is  the  United  States  legal  value  of  the  above  ? 

Ans.  $1210. 

3.  Reduce  1£  sterling  to  Federal  money  at  9^%  advance  upon 
the  nominal  value.  Ans.  $4.86|. 

4.  Reduce  40£.  10s.  to  Federal  money,  at  9^  %  advance.  (See 
Art.  265,  Note.) 

5.  What  will  be  the  value  in  Federal  money  of  84£.  19s.  ll|d. 
at  10%  advance? 

6.  What  is  the  cost  in  New  York  of  a  bill  on  Liverpool  for 
470£.  13s.  9d.,  at  9|%  premium? 

7.  A  gas  company  purchased  Newcastle  coal  in  England  to 
the  amount  of  1000£.  15s.  7f  d. ;  exchange  being  8^%  premium, 
what  will  this  be  worth  in  United  States  money  ? 

8.  A  dealer  in  flour  shipped  to  London  3000  barrels  of  flour, 
which  cost  $4.20  a  barrel  in  Baltimore ;  the  flour  was  sold  in 
London  at  1£.  Gs.  6d.  per  barrel, exchange  being  10%  advance; 
what  was  the  gain,  without  regard  to  expenses  ? 

9*  How  much  Federal  money  will  pay  for  3  T.  15  cwt.  2qr. 
1  lb.  of  iron,  at  7£.  10s.  9d.  sterling  per  ton,  when  the  premium 
is  9^%? 
-<10!  A  cotton  broker  sent  to  Manchester,  England,  50  bales  of 


EXCHANGE.  235 

cotton  averaging  460  pounds  each;  the  cotton  was  sold  at  lid. 
per  pound.  What  was  the  amount  of  sales  in  United  States 
money,  premium  being  lOf  %,  and  what  was  the  broker's  com- 
aaission  at  l^fo  upon  the  sales?  jf     ,9    /  i^ 

11.  Reduce  $200  to  sterling  money  when  exchange  on  Eng- 
land is  10%  advance.         200  x  9  x  100  _  40£.  I8s.  2^yd.,  Ans, 

/''  40X110  11? 

12.  Reduce  $575  to  sterling  money  at  9f  %  advance. 

13.  When  exchange  is  9  5^  premium,  what  is  the  value  of 
$6874.40  ? 

14?  What  will  a  merchant  gain  by  buying  4000  bushels  of 
corn  in  Albany  at  65  cents  per  bushel,  paying  for  shipping  and 
other  expenses  30  cents  per  bushel,  and  selling  it  in  Liverpool, 
England,  at  4s.  9d.per  bushel,  when  exchange  is  10%  premium? 

15!  A  merchant  in  1864  shipped  to  Liverpool  5000  pounds  of    '' 
butter,  which  cost  him  in  New  York  35  cents  per  pound;  he"'*"'' 
paid  4fo  for  freight  and  other  expenses,  and  sold  it  in  Liverpool   ^ 
for  lid.  per  pound.     Exchange  being  120%  premium,  did  he    '^ 
gain  or  lose,  and  how  much  ?  v 

16.  What  is  the  cost  of  a  set  of  exchange  on  Paris,  for  6OOO  ^^ 
francs,  exchange  being  6|  francs  per  dollar  ? 

17.  What  is  the  cost  of  a  set  of  exchange  on  Paris  for  4500  ^     j 
francs  at  5%  premium,  the  value  of  1  franc  being  18|  cents ?^^^  'K   f  ^^ 
'/■  18.  Mr.  James  Payne,  an  American  gentleman,  while  trav- 
elling in  England  received  the  following  draft;  what  was  the  cost 

of  the  draft  in  America,  the  premium  being  41  %  in  favor  of 
England  ? 

£127  Boston,  Au^.  23,  1863. 

At  sight  of  this,  our  jirst  Bill  of  Exchange  (second  and  third 
of  same  tenor  and  date  unpaid),  pay  to  the  order  of  James  Payne, 
in  Manchester,  one  hundred  and  twenty-seven  pounds  sterling^ 
value  received,  and  charge  the  same  to  our  account. 

J.  E.  Thayer  &  Bro.  ^) 

To  Messrs.  Calmont  Bros.  &  Co,  Bond  Street,  London,     j  j  ^v    1^  ^£  <f  J 

5^  For  Dictation  Exercises,  see  Key. 


236  PEllCEi<TAd?. 

3«>3«     Questions   in   Heview. 

"What  is  Percentacje  *  From  what  is  per  cent,  derived  ?  In  what 
four  ways  represented?     Represent  |^  decimally. 

How  is  a  per  cent,  reduced  to  lowest  terms  ? 

How  is  a  common  fraction  reduced  to  a  per  cent.  ? 

What  is  the  complement  of  any  rate  per  cent.  ? 

Reduce  |  to  a  per  cent. ;  represent  it  decimally ;  find  its  comple- 
ment, and  reduce  that  to  its  lowest  terms. 

How  do  you  find  any  required  per  cent,  of  a  number  ? 

Name  the  applications  of  percentage  in  this  book.     (See  Contents.] 

Upon  what  is  the  percentage  of  Profit  or  Loss  reckoned  ? 

If  goods  cost  24  cents,  for  what  must  they  sell  to  gain  8^^  ?  to  lose 
1 6f ^  ?  What  per  cent,  would  be  gained  or  lost  by  selling  the  above 
at  30  cents  ?  at  21  cents  ?  If  24  cents  is  20^  less  than  the  value  of 
goods,  what  is  the  value  ?  if  24  cents  is  33^^  more  than  value  ?  If 
18  cents  is  10^  less  than  cost,  for  what  would  you  sell  to  gain  10^  t 
to  lose  25^  ?  If  10^  of  what  you  receive  for  goods  is  gain,  what  is 
your  gain  per  cent.  ? 

What  is  Interest  ?  principal  ?  amount  ?  legal  rate  ?  usury  ? 

Usual  rate  in  United  States?  Rule  for  finding  interest  on  $1  a1 
6^  ?  on  any  sum  ? 

At  6%  in  what  time  will  a  sum  double  ?  will  the  interest  equal  ^ 
of  the  principal?  J?  ^?  1?  i?  i?  ^?  tV?  l^?  ^?  ^V?  i^  ?  ToW? 

At  6^,  what  part  of  the  principal  is  the  interest  for  1  mo.  ?  foi 
3  mo.  ?  5  mo.  ?  6f  mo.  ?  11  mo.  ?  13  mo.  10  d.  ?  1  y.  4  mo.  20  d.  ?  1  y. 
8  mo.  ?  5  y.  6  mo.  20  d.  ? 

What  is  the  interest  on  $1  for  3  y.  4  m.  ?  6  d.  ?  3  d.  ?  1  d.  ?  20  d.J 
33 J  mo.  ? 

How  obtain  interest  at  any  rate  besides  6^  ? 

How  obtain  interest  on  pounds,  shillings,  &c.  ? 

What  is  the  rule  for  reducing  shillings,  pence,  and  farthings,  to 
decimal  of  l£  by  inspection  ?  What  is  the  value  of  Is.  in  decimal  of 
£1?  3s.?  OS.?  Iqr.?  12  qr.?  6d.  ?  9d.? 

Name  the  first  month  in  the  year ;  the  fourth ;  the  seventh ;  the 
tenth  ;  third  ;  twelfth  ;  ninth ;  fifth  ;  eighth. 

How  many  months  and  days  forward  from  June  7th  to  the  2d  of  each 
of  the  above  ?  from  November  15th  ?  from  March  28th  ? 

What  are  Partial  Payments  ?  AVTiat  is  the  legal  rule  fcr  partial 
payments  ?     How  is  the  record  of  payments  kept  ?     What  are  the  pay 


r.UESTIONS  IN  REVIEW.  237 

jnts  called?     Suppose  an  endorsement  will  not  cancel  the  interest? 
;  lie  in  common  nse  when  the  note  is  paid  within  one  year  ?     Rule  for 
nual  interest  ? 

What  is  Compound  Interest  ?  How  often  may  interest  be  com- 
•unded  ?     For  how  many  periods  of  time  will  interest  be  compounded 

2  y.  9  mo.,  if  it  is  compounded  semi-annually  ?  quarterly  ?  monthly  ? 
What  three  factors  are  employed  to  produce  interest  ?  The  inter- 
t,  principal,  and  rate  being  known,  give  the  rule  to  find  time.  The 
terest,  principal,  and  time  being  known,  give  the  rule  to  find  rate.  The 
terest,  rate,  and  time  being  known,  give  the  rule  to  find  principal, 
/"hat  is  the  dividend  in  each  case  ?  To  find  time,  for  what  time  do 
)u  take  interest  for  a  divisor  ?  to  find  rate,  at  what  rate  ?  to  find 
•incipal,  on  what  principal  ?  Amount,  rate,  and  time  being  known, 
ve  rule  to  find  principal. 

What  does  Present  Worth  embrace  ?     What  is  discount  ?     Give 
de  for  present  w^orth.     How  find  discount  ?     How  prove  the  work  ? 
What  is  a  Promissory  Note  ?    What  the  face  of  a  note  ?    What 
bank  discount  ?     What  are  days  of  grace  ?  avails  of  a  note  ? 
Which  is  the  larger,  true  or  bank  present  worth?  true  or  bank 
iscount  ? 

Describe  the  process  of  getting  a  note  discounted  at  a  bank. 
What  is  endorsing  a  note  ? 

Rule  for  finding  the  face  of  a  note,  which,  discounted  at  a  bank, 
ill  yield  a  certain  sum  ? 

What  is  Commission  ?  Brokerage  ?  What  are  Stocks  ?  When  are 
ocks  above  par  ?  below  par  ?  at  an  advance  ?  discount  ?  premium  ? 
Upon  what  is  the  per  cent,  of  commission  or  brokerage  estimated  ? 
What  is  the  rule  for  finding  what  sum  is  to  be  laid  out  when  a  re- 
ittance  is  made  which  contains  that  sum  together  with  the  commis- 
on  ?     How  obtain  commission  ? 

What  is  Insurance  ?  policy  ?  premium  ?  W^hat  are  underwriters  ? 
What  is  the  rule  for  finding  the  policy  which  will  cover  a  certain 
im  and  the  premium  ? 

What  is  Equation  of  Payments?  the  interest  rule?  Pvule  for 
juating  an  account  ? 

What  are  Taxes  ?  polls  ?  real  estate  ?  personal  property  ?  assessors  ? 
:ow  find  tax  on  $1  ? 

What  are  Custom-houses  ?  Duties  ?  What  is  a  specific  duty  ?  an 
i  valorum  duty  ?  leakage  and  breakage  ?  tare  ?  tonnage  of  vessai^  ? 
f08s  weight  ?  net  weight  ? 


238  PERCENTAGE.  j 

What  is  a  Draft  ?  Who  is  the  drawer  ?  the  drawee  ?  the  payee  i 
the  holder  ?     What  is  the  endorsement  of  a  draft  ?  the  acceptance  ? 

What  is  a  Bill  of  Exchange  ?  What  is  Inland  Exchange  ?  For- 
eign Exchange  ?  When  a  bill  negotiable  ?  when  at  a  premium  ?  when 
at  a  discount  ?  Define  course  of  exchange  ;  a  set  of  exchange.  Give 
the  rule  for  finding  the  value  of  a  draft ;  for  finding  the  face  of  a 
draft. 

What  is  the  nominal  value  of  1£  sterling?  the  legal  value?  the 
intrinsic  value  ? 

Upon  what  is  Sterling  Exchange  reckoned  ? 

3^4.     Miscellaneous  Examples. 

1.  Reduce  -j-V  to  a  per  cent. 

2.  Represent  Ig^'g^  decimally.  ^ 

3.  Reduce  106^^  to  its  lowest  terms.  ^' 

4.  Find  the  complement  of  84^%. 

5.  What  is  124%  of  5  T.  3  cwt.? 

6.  What  is  the  amount  at  6%,  simple  interest,  of  |o8.75,  from 
Aug.  5  to  Nov.  10  ? 

7.  What  is  the  amount  of  $380.25,  at  6%  compound  interest, 
for  2  yrs.  5  mo.  ? 

8.  What  is  the  simple  interest,  at  5%,  of  10£.  8s.  6d.,  for  4  yrs. 
2  mo.  ? 

9.  What  is  the  compound  interest  of  the  above  at  the  same 
rate  and  for  the  same  time  ? 


10.  $2500.  Chicago,  April  4,  1862. 
In  three  months  from  date,  I  promise  to  pay  John  Peirce, 

or  bearer,  twenty-five  hundred  dollars,  with  interest  after  at  6^, 
value  received.  P.  T.  Ivison. 

The  above  was  endorsed  as  follows  :  $1010  March  28,  1863; 
$100  Aug.  10,  1864;  $1000  Jan.  1,  1865.  What  was  still 
due? 

11.  At  what  per  cent,  must  $450  be  on  interest  9  months  to 
gain  $13.50? 

12.  I  received  $7.35  for  the  use  of  $150  a  certain  time  at  7%. 
Required  the  time. 


MISCELLANEOUS  EXAMPLES.  239 

13.  Lent  a  certain  sum  for  1  y.  6  mo.  at  5%  ;  the  interest  be- 
ing |>9.40,  What  was  the  sum?     j     ?  /,  ^      -■) 

14.  What  principal  will  amount  to  ^63.25  in  1  y.  3  mo.  at 
8%? 


15.  $150.25.  WiNTiiROP,  Jan.  5,  1860. 

On  the  fifteenth  of  May,  1860, 1  promise  to  pay  to  the  order  of 
B.  F.  Tweed  one  hundred  fifty  ^J'q  dollars,  with  interest  after, 
at  6%,  value  received.  D.  D.  Daniels. 

If  the  holder  of  the  above  have  it  discounted  at  a  bank  Feb. 
15,  1860,  what  will  he  receive  ? 

1 6.  "What  will  be  the  true  present  worth  of  the  above  at  its 
date?  /-<y/\-   k 

17.  What  would  settle  the  above  Oct.  27,  1860  ? 

18.  What  would  settle  the  above  at  compound  interest  Oct. 
27,  1865? 

19.  What  must  be  the  face  of  a  note,  which,  discounted  at  a    , 
bank  for  30  days  and  grace,  would  yield  8500  ?  ^,  0\  j  -h 

20.  In  what  time  will  a  sum  of  money  double  at  2  5^  simple 
interest  ?  J^/^v  Q 

21.  Find  the  commission  for  selling  the  following  lot  of  hogs 
at  ^%l 

Sales  on  %  of  Messrs.  Bishop  &  Trowbridge, 
By  BoNNEY  &  Waite. 


7  Hogs,  as  follows:  293  lb.  317  lb.   300  lb. 

219  lb.  314  lb.  323  lb.  184  lb. 

Amount, at  7 J-  cts.  per  lb. 

22.  What  will  a  broker's  bill  be  for  5  shares  of  stock  purchased 
for  me  at  7  %  advance,  shares  having  originally  been  $500,  his 
brokerage  at  ^%  included  ?     <    i^    V  V      j    )'  -{  x, 

23.  Required  the  value  of  1£  sterling  at  9%  advance  upon 
the  nominal  value. 

—24.  What  will  be  the  contents  of  a  piece  of  cloth  originally 


240  PERCENTAGE. 

1  yard  long  by  2^  yards  wide,  after  sponging,  if  in  that  operation 
it  shrinks  4%  in  length  and  6%  in  width  ? 

25.  A  commission  merchant  receives  $544 ;  of  this  he  is  to 
invest  such  a  portion  as  remains  after  deducting  his  commission 
of  2^%.     What  is  his  commission,  and  what  will  remain  ?  U? 'v.^  / 1 

26.  What  is  the  cost  of  insuring  $2500  at  $17.50  on  |1000? 

27.  What  will  be  the  net  loss  to  an  insurance  company  in  case 
of  the  loss  by  fire  of  a  property  insured  for  $4500,  on  which  the 
company  had  received  3%  premium,  no  allowance  for  interest?'^/  j 

28.  For  what  sum  must  a  policy  be  taken  out  to  cover  $2575 
and  the  premium  of  1|^  ? 

29.  What  per  cent,  of  10£  is  15s.? 

30.  Of  how  many  rods  is  84  rods  87^-%  ? 

31.  Take  9%  of  7.5  acres. 

32.  Paid  $18.77  for  insuring  my  schooner  at  a  premium  of 
^'Jo  ;  what  was  the  sum  covered? 

33.  What  is  the  par  value  of  stock,  which,  selling  at  25^ 
above  par,  brings  $500  ? 

34.  In  the  year  1862  the  town  of  B  voted  to  raise,  by  taxes, 
$97290 ;  -^  of  Miis  was  levied  upon  the  polls ;  the  valuation  of 
the  town  w^'^  ^10134375 ;  what  was  the  tax  on  $1,  and  what 
was  the  ta>^  f^*  "x  non-resident  who  owned  a  house  in  town  valued 
at  $2000  ? 

35.  E.f?4rce  750£.  10s.  4d.  sterling  to  United  States  currency, 
exchange  at  17%  advance  upon  the  nominal  value,  i       _^  !^^"\\  J 

36.  A  debtor  owes  $200,  ^  due  in  2  months,  ^  in  3  months, 
and  the  remainder  in  5  months ;  what  is  the  equated  time  for 
paying  the  whole  ?     ; 

37.  If  ^  of  a  sum  of  money  be  due  in  2  months,  |-  in  4  months, 
^  in  3  months,  and  the  remainder  in  4  months,  in  what  time 
should  the  whole  be  paid  ? 

38.  What  is  the  average  time  for  paying  for  $200  worth  of 
goods  purchased  May  17,  1859,  on  4  months'  credit;  $500  worth, 
purchased  June  18,  1859,  on  60  days'  credit ;  and  $300  Avorth, 
purchased  June  19,  1859,  on  90  days'  credit? 

.  89.  There  is  a  note,  dated  July  1,  on  60  days'  credit,  for  $200. 


EXAMPLES  IN  PllOm^  AND  LOSS. 


24!. 


July  20  there  is  paid  $50;  Aug.  19,  $60;  and  Aug,  21,  $10. 
When  should  the  balance  be  paid  ? 

40*  Thomas  Swain  owes  Wm.  C.  Chapin  $500,18,  due  Jan,  7, 
1863,  and  $207.06,  due  April  4,  1863;  Mr.  Chapin  owes  Mr. 
Swain  $800,  due  Feb.  S,  1863.  When  should  the  balance  be 
paid? 

4lt  What  is  the  cash  value  of  the  above  March  25,  1863  ? 

42f  What  is  the  date  of  a  note  which  must  be  given  to  settle 
the  following,  allowing  4  months'  credit  on  each  of  the  merckan" 
dise  items? 


John  F,  Stone  in  % 

with  Thomas  Emerson's  Sons. 


Dr.       Cr. 


1863. 
Sept.  15. 
Dec.     1. 

1864. 
Feb.  10. 
Mar.  1. 
Jan.  9. 
"  25. 
Apr.      1. 


To  200  bbk.  Apples,  (a)  $3.25, 
100    *'      Flour,     «     3.48, 


"  35,000  Bricks, 
«  Cash,  .  .  ^ 
By  Merchandise,   , 

do., 
"    Cash,  .     .     . 


6.00,  per  M. 


650 


300 


00 


00 


510 
476 
265 


Do^S. 


3^^*     Miscellaneous  Examples  in  3?rofit^and 

Note.  —  Younger  pupils  can  omit  this  article  till  they  review. 

1.  By  selling  goods  at  6  cents  3  mills  per  yard,  I  lose  30%  ; 
what  did  they  cost  ? 

2.  If  I  lose   10%  bj  selling  goods  at  18  cents  per  yard,  for 
what  should  they  have  been  sold  to  gain  20  %  ? 


18X100 


=  20  cts.i  the  cost  : 


20  X  I'-iO 


zzz  24  cts.      Ans, 


JW  100 

3!  By  selling  a  lot  of  iron  at  12^%  below  cost,  I  received 
$14.75  less  than  I  should  have  received  if  I  had  sold  it  at 
12^%  above  cost;  what  did  it  cost?  what  should  it  have  sold 
for  to  gain  12^%  ? 

4.  A  merchant  bought  5  cwt,  1  qr.  of  coifcc  for  $03  ;  for 
wh.iit  must  he  sell  it  per  lb.  to  gain  16§%  ? 

5.  For  Avhat  must  hay  be  sold  per  ton  to  gain  13^-%,  if  by 
selling  at  $16,  33^%  be  gained? 

Note.— $16  -5-  1.33^  =  |5l2;    1,13^  of  $12  =  $13,62,  Anu 
16 


242  SCPERCENTAGE. 


/! 


6.  If  12^%  bo  gained  by  selling  ladies'  slippers,  at  $9  per 
doze^i  pairs,  for  what  should  they  have  been  sold  per  pair  to  gain 
oD%?  Ans.  90  cents. 

7.  13%  is  lost  by  selling  a  lot  of  land  for  $783  ;  what  would 
it  have  brought  if  it  had  been  sold  at  a  loss  of  8^%  ? 

A71S.  S825. 

8.  50%  of  a  certain  number  exceeds  35%  of  it  by  $13.70  ; 
what  is  the  number?  Ans,  $91.33^. 

9.  A  person  takes  a  note  on  2  months'  credit  for  $110  in  pay- 
ment for  a  watch ;  on  getting  the  note  discounted  at  a  bank,  he 
iinds  that  he  has  lost  40  %  on  the  first  cost  of  the  watch.  Re- 
quired the  cost  ?  Ans.  $181.40f. 

10.  A  broker  purchases  a  lot  of  stocks  at  an  average  of  9% 
below  par,  and  sells  them  at  an  average  of  7f  %  above  par,  and 
makes  $300  ;  what  was  the  par  value  of  the  stocks  ?  /  (         ; 

11.  If, by  selling  goods  at  60  cents  per  lb.,  20%  is  gained,  what 
%  would  have  been  gained  by  selling  them  at  75  cents  per  lb.  ? 

Ans.  50%. 

12.  If  10%  is  lost  by  selling  boards  at  $7.20  per  M.,  what  % 
would  be  gained  by  selling  them  at  90  cents  per  C.  .'*  1 

13.  Sold  boots  at  $3.55  per  pair,  and  thereby  lost  29%  ;  what 
%  would  have  been  lost  by  selling  them  at  $57.50  per  dozen 
pairs  ? 

14.  A  dealer  has  18  barrels  of  sound  apples  remaining  in  a  lot 
of  which  10%  have  decayed;  if  his  lot  cof.t  him  $1.50  per  bbl.. 
would  he  gain  or  los3  on  the  lot,  and  what  %  by  selling  the  re- 
mainder at  $1.75  per  bbl.  ?  Ans.  5  %  gain. . 

15.  Sold  4  ploughs  at  $24  each ;  on  2  of  them  I  made  20%, 
and  on  2  I  lost  20%;  what  did  I  gain  or  lose  on  the  whole  ? 

Ans.  Lost  $4.00. 

16.  20%  of  a  lot  of  barley,  originally  5000  bushels,  was  de- 
stroyed by  fire,  the  cost  having  been  $1^  per  bushel ;  what  per 
cent,  will  be  gained  on  the  lot  by  selling  the  remainder  at  $2  per 
bushel  ? 

17.  By  losing  3  cents  on  a  pound,  I  get  87|-%  of  the  cost  of 
butter ;  if  I  had  lost  4  cents  a  pound,  what  %  should  I  have 
receivecl.^  J[w5.  83]%. 


EXAMPLES  IN  PROFIT  AND  LOSS.  24S 

18.  By  having  5  pupils  absent  from  school  my  attendance  is 
93|%;  if  my  attendance  had  been  95^,  hoAV  many  pupils  must 
have  been  absent  ?  Ajis.  4  pupils. 

19.  What  will  be  the  %of  gain  on  the  cost  of  174  shares  of 
Roxbury  Gas  Co.'s  stock,  nominal  value  of  shares  being  $87.50, 

if  it  be  bought  at  15%  below  par,  and  sold  at  19^%  above  par?  ^   > 

20.  If  I  buy  coal  at  $4.12  per  ton  on  6  months'  credit,  for 
what  must  I  sell  it  immediately  to  gain  10^^?  Ans.  $4.40. 

21.  If  I  pay  $3.90  cash,  for  what  must  I  sell  it  on  4  months  to 
gain  20%  ? 

22.  If,  by  .>clling  wood  at  75  cts.  per  cd.  ft.,  6|%  be  lost,  for 
what  should  it  have  been  sold  per  cord  to  gain  34%?  Ans.  $6,624. 

23.  I  sell  ^  of  a  lot  of  goods  for  $9,  and  thereby  lose  25%; 
for  what  must  I  sell  the  remainder  to  make  8^% on  the  whole  ? 

Ans.  $30. 

24.  A  grocer  bought  100  gallons  of  oil,  at  $1  per  gallon  ;  he 
mixed  with  it  50  gallons  that  cost  $1.60  per  gallon,  then  sold  the        /7^ 
whole  at  the  rate  of  $1.44  per  gallon ;  did  he  gain  or  lose,  and        ^L 
what  %  ?    §   :  ^ 

25.  Suppose  the  above  was  sold  on  a  credit  of  6  months,  what 
was  the  %  of  gain? 

26.  Suppose  the  oil  to  have  been  bought  on  4  months,  and  sold 
for  cash ;  what  %  was  gained  ?  ^S^X  ~^ 

27.  For  what  should  he  sell  the  above  per  gallon  to  make  2a 
%,  if  he  bought  for  cash  and  waited  10  months  for  his  pay  f  "  ; 

28.  How  much  water  must  be  mixed  with  a  barrel  of  ink  (31 
galls.),  which  cost  $34.10,  that  it  may  be  sold  at  $1.10  a  gallon, 
and  a  profit  of  25%  be  realized  by  it?  Aiis.  7|  gall, 

29.  What  water  would  be  required  in  the  above,  if  the  cost 
had  been  $25,  the  profit  20  %  ,  and  the  selling  price  $.75  per 
gallon  ? 

30.  If  20%  of  what  I  receive  for  an  article  is  gain,  what  is 
the  gain  %  ? 

Note.  —  If  20j^  is  gain,  80^  is  cost;  the  gain  then  ^  ^  of  the  cost, 
which  equals  25^,  Ans. 

31.  If  25%  of  what  I  receive  is  gain,  what  is  the  gain  %  ?, 


^o 


244  PERCENTAGE. 

32.  When  two  sovereigns  are  sold  for  124.20,  what  is  the 
premium  on  gold,  estimating  the  sovereign  at  its  legal  value  ?  j  Jy/} 

Bo.  What  amount  of  current  money  must  a  broker  give  me 
for  120  in  gold,  when  the  premium  on  gold  is  178%  ?    TT,P  ^ 

34.  A  man  hands  a  broker  $40  in  currency,  saying,  "  Give  me 
the  value  of  this  in  gold ; "  gold  is  "  quoted "  at  $2.64 ;  how 
many  dollars  did  the  broker  return,  and  how  much  change  in 
currency  ? 

35.  Suppose  the  broker  paid  20%  premium  for  this  gold,  what 
did  he  gain  upon  the  $15,  no  allowance  for  interest  ?  4  S/^'^ 

36.  What  %  did  he  gain  on  the  money  he  expended  to  pur- 
chase the  gold  ?       I  ^  '  ^- 

37.  A  man  about  to  travel  in  Canada  bought  $50  in  gold, 
when  gold  was  at  a  premium  of  150%  ;  what  amount  of  current 
money  did  he  pay  for  it  ?    /  2  0\  ^ 

38.  When  gold  is  at  a  premium  of  50%,  what  is  the  depreci- 
ation in  the  currency  ? 

39.  A  merchant  imports  from  Hamburg  a  bale  of  cloth,  con- 
taining 12  pieces,  40  yards  each  ;  the  cloth,  with  charges  there, 
cost  him  $480  ;  he  pays  a  duty  here  of  35  cts.  per  yd.,  freight 
$28.50,  and  other  charges  $7.11 ;  at  what  must  he  sell  the  cloth 
per  yd.  to  gain  10  %  above  all  charges  ?    /  M  ^-(^  ^  -j^ 

40.*  Vinton  &  Morris  imported  a  lot  of  W.  I.  sugar,  for  which 
they  paid  $15272  in  Havana;  their  expenses  amounted  to  5% 
of  the  first  cost;  10  months  after,  they  sold  it  so  as  to  gain 
$5S12.46  above  all  expenses,  including  interest  at  6%;  what  % 
Aid  they  gain  ?    f  '    - \^ 

41.  An  importer  received  from  Holland  75  gross  quart  bottles 
of  beer,  invoiced  at  10  cts.  per  bottle  ;  a  duty  of  30  cts.  per  gal. 
was  paid  at  the  custom  house  after  10%  had  been  deducted  for 
breakage ;  if  all  the  bottles  be  found  sound,  what  %  will  be 
gained  by  selling  the  beer  at  25  cents  per  bottle  ? 

3^6.    General  Revievt,  No.  7. 

1.  What  is  }  per  cent,  of  $56.49  ?.^  ' 

2.  What  is  the  amount  of  $84.20  for  3  yrs.  5  mo.  12  ds.,  at  5 
per  cent.  ?     1  '^ .   (   '  0 


GENERAL  REVIEV/.  245 

3.  What  is  the  interest  of  24  £.  7  s.  6  d.  for  2  yrs.  4  mo.  ? 

4.  Oct.  12,  1861,  gave  my  note  on  demand,  with  interest,  for 
$480.  Feb.  6,  1862,  paid  $120.  What  remained  due  Aug.  24, 
1862? 

5.  I  held  a  note  for  $500,  which  bore  interest  from  May  10, 
1859.  Sept.  16,  1860,  received  $140;  July  28,  1862,  received 
$50.     What  remained  due  Sept.  4,  1862  ?  ^1),)^^ 

6.  If  I  pay  $45  interest  for  the  use  of  $500  for  3  years,  what 
is  the  rate  per  cent  ?   0  3 

7.  How  long  must  $204  be  on  interest  at  7  per  cent,  to 
amount  to  $217.09?  I  I  r<y^,r>^b 

8.  What  principal  will  gain  $9.20  in  4  mo.  18  ds.,  at  4  per 
cent.  ? 

9.  What  sum,  at  7  per  cent.,  will  amount  to  $221,075  in  3 
yrs.  4  m.? 

10.  What  is  the  compound  interest  of  $600  for  1  yr.  4  mo., 
interest  payable  semi-annually  ?     /  V ,  w 

11.  What  is  the  present  worth  of  a  note  for  $488.50,  due  in  2 
yrs.  5  mo.  15  ds.,  at  9  per  cent.? 

12.  What  is  the  discount  of  $105.71,  due  4  yrs.  hence  ? 

13.  What  commission  must  be  paid  for  collecting  $17380,  at 
3^  per  cent.  ? 

14.  What  amount  of  stock  can  be  bought  for  $9682,  and 
allow  \  per  cent,  brokerage? 

15.  What  is  the  value  of  20  shares  bank  stock,  at  8^  per  cent. 
discount,  the  par  value  of  each  share  being  $150  ?  ^  /  ^  S^f 

16.  What  sum  will  be  received  from  a  bank  for  a  note  of 
1260,  payable  in  4  months  ?  -;  ^s  ^\  \  I  \  ""^ 

17.  What  is  the  bank  discount  on  $320  for  90  days  ? 

18.  What  is  the  face  of  a  note  which  yields  $112,803,  when 
discounted  at  a  bank  for  60  days  ?    \\^\\ 

1 9.  A  house,  valued  at  $4750,  is  insured  at  J  of  1  per  cent. ; 
what  is  the  premium  ?     j  /;    i.  J  lyO 

20.  What  is  the  duty,  at  l5  per  cent,  ad  valorem,  on  20  bags 
of  coffee,  each  containing  115  lbs.,  valued  at  42  cts.  per  lb.?  y//i/ 


?46  EATIO. 


RATIO. 


Scir*  Hatio  is  the  relation  which  one  number  bears  to 
a  lother  number  of  the  same  kind. 

Ratios  are  of  two  kinds,  Arithmetical  and  Geometrical. 

3^80  Arithmetical  Ratio  is  ratio  of  numbers  with  respect 
to  their  difference  ;  as  6  —  4=2. 

GEOMETRICAL  RATIO. 
359.     Geometrical  Ratio  is  ratio  of  numbers  with  respect 
to  their  quotient ;  as  2  : 4  =  4,  read  2  is  to  4,  or  the  ratio  of  2 
to  4  =:  J ;  6:3  =  2,  read  6  is  to  3,  or  the  ratio  of  6  to  3  =  2. 

300.  The  first  term  of  a  ratio  is  called  the  Antecedent, 
the  second,  the  Consequent ;  both  together  are  called  a  Coup- 
let. 

What  is  the  antecedent  in  the  first  illustration  in  Article  359  ?  the 
consequent  in  the  second  ?  the  ratio  in  the  first  ?  the  consequent  in  the 
first?  the  ratio  in  the  second  ? 

301.  When  the  terms  of  a  ratio  are  equal,  the  ratio  is  one 
of  equality;  when  the  antecedent  is  greater  than  the  conse- 
quent, it  is  a  ratio  of  greater  inequality ;  when  the  antecedent 
is  less  than  the  consequent,  it  is  a  ratio  of  less  inequality. 

36^«     It  will  be  readily  seen  that  ratios,  being  expressions 
for  division,  are  similar  to  fractions.     They  can  at  any  time  be  ■ 
written  in  a  fractional  form,  the  antecedent  taking  the  place  of 
the  numerator,  and  the  consequent  that  of  the  denominator.    The 
principles  applicable  to  fractions  apply  also  to  ratio.     Hence, 

Multiplying  the  antecedent,      )        ,  .  ,. 

7.  .7.       ,7  ,  V  multiplies  the  ratio. 

or  dividing  the  consequent,  )  ^ 


Dividing  the  antecedent,  t    7.   . , 

^  divides  the  ratio. 


or  multiplying  the  consequent,  ) 
Multiplying  or  dividing  both  ter 
of  a  ratio  by  the  same  number y 


Multiplying  or  dividing  both  terms )    .  7       .         , 

r  does  not  alter  its  value. 


GEOMETRICAL  RATIO.  247 

363.  Ratios,  like  fractions,  may  be  simple,  complex,  or 
compoimd.  A  ratio  is  sim.ple  when  each  term  is  a  simple 
number;  it  is  complex  when  either  term  contains  a  fraction  ;  it 
is  compound  when  it  is  the  indicated  product  of  two  or  more 
ratios. 

Simple  Ratio.  Complex  Katie.  Compound  Eatio. 

2     24 
2:8.  j--f,  2X3:5X5. 

364:,     Exercises. 

1.  Write  the  ratio  of  2  to  3;  of  7  to  10 ;  of  f  to  f ;  of  2  X  7 
to  5  X  4. 

2.  Multiply  the  ratio  o  :  4  by  2. 

3.  Divide  the  same  by  2. 

4.  Reduce  the  ratio  6  :  6  to  lower  terms. 

5.  Write  any  ratio  of  equality  ;  of  greater  inequality ;  of  less 

inequality. 

5 
300.     III.  Ex.     Reduce  f  :  —  to  a  simple  ratio. 


^^ 


Opekatiox. 
-U .    Multiplying  each  term  of  the  ratio  | :  J^^-  by  3  X  7,  we 


have :  — r=14:45,  Ans.     Hence, 

jo  A 

To  reduce  a  complex  ratio  to  a  simple  one :  Reduce  each  term 
to  its  simplest  for  m^  then  multiply  each  b^  the  least  common  multi- 
ple of  the  denominators,  and  cancel. 

Reduce  to  simple  ratios, 

1.  2x  =  l. 

2.  ^  =  8. 


3. 

3X5:4X8. 

4. 

5x4:3Xf 

5. 

|X5:|X1| 

248  PKOPORTION. 


PROPORTION. 


360:)  Proportion  is  an  expression  of  equality  between  two 
ratios ;  thus,  2:3  =  4:6,  read  2  is  to  three  as  4  is  to  6 ;  that 
is,  2  is  the  same  part  of  3  that  4  is  of  6.  2  is  f  of  3,  and  4  is  § 
of  6. 

36T,  The  first  and  fourth  terms  of  a  proportion  are  called 
the  extremes,  and  the  second  and  third  are  called  the  means. 
The  first  ratio  is  called  the  first  couplet,  and  the  second  ratio 
the  .second  couplet.     Read  the  following  proportions:  — 


1. 

5 :  10  =z  S :  6. 

a 

8:2  =  12:3 

2. 

^:lz=:5:15. 

4. 

06:7  =  8:1 

Name  the  extremes  of  the  first  proportion;  the  means  of  the  second; 
the  antecedents  of  the  third ;  the  consequents  of  the  fourth  j  the  sec- 
ond couplet  of  the  first  proportion. 

308.  Inverse  Proportion.  Four  terms  are  directly 
proportional  when  the  first  is  to  the  second  as  the  third  is  to 
the  fourth.  They  are  inversely  proportional  when  the  first 
is  to  the  second  as  the  fourth  is  to  the  third,  or  when  one  ratio  is 
direct  and  the  other  inverse.  Thus,  the  amount  of  work  done  in 
any  given  time  is  directly  proportional  to  the  men  employed ;  i.  e.> 
the  more  men,  the  more  work ;  but  the?  time  occupied  in  doing  a 
certain  work  is  inversely  proportional  to  the  men  employed ;  i^  e., 
the  more  men,  the  less  time. 

30O«  A  compound  proportion  is  an  equality  between  a 
compound  ratio  and  a  simple  ratio,  or  between  two  compound 
ratios. 

STO*  Three  terms  are  in  proportion  when  the  first  is  to  the 
second  as  the  second  is  to  the  third.  The  second  term  is  called 
a  mean  proportional  between  the  other  two ;  thus,  in  the  pro- 
portion, 3:6=:6:12,  6isa  mean  pix)portional  between  3  and  12. 

3TI*  The  performance  of  arithmetical  examples  by  pro 
portion  depends  upon  the  follow hig  important  principle:  — 


SIMPLE  PROPORTION.  249 

In  every  proportion  the  product  of  the  means  equals  the  product 
of  the  extremes. 

Illustration.  Writing  the  given  proportion  in  a  frae- 

Q  .  o 4-6  tionai  form,  we  have  f  =:  |.    Muhiplying 

2 A  each  fraction  by  the  product  of  the  de- 

^        ^                 .  nominators,  and  cancelling,  we  have  2  X 

^  X  3  X6__  4X3X0  6  z=  4  X  3.     But  2  and  6  are  the  extremes, 

3                         0  and  4  and  3  the  means ;  hence  the  product 

2X6  =  4X3  of  the  extremes  equals  that  of  the  means. 

373,  From  the  abovo,  it  follows  that  whenever  an  extreme 
in  a  proportion  is  wanting,  it  can  be  found  hy  dividing  the  product 
of  the  7neans  by  the  given  extreme  ;  and  whenever  a  mean  is  want- 
ing, it  may  he  found  hy  dividing  the  product  of  the  extremes  hy  the 
given  mean. 

Supply  the  terms  wanting  in  the  following  proportions:  — 


1.         5:1  —  50:? 

3.         9:7rr:?:5G. 

2.         8  :  ?  =  3  :  10. 

4.         ?  :  2  =:  15  :  5. 

37S.     In  the  proportion  2 

:  4  ==  4  :  8,  42  z=  2  X  ^ 

-.4=:= 

V2  X  8  (Arts.  91,  92) ;  hence  a  mean  proportional  between  twd 
nmnbers  equals  the  square  root  of  their  product. 

Supply  the  mean  proportionals  between  the  following  numbers 
and  w^rlte  the  proportions  :  — 


I        7.     2  and  24^. 
6.     2  and  18.        I        8.     20  and  5. 


9.     3  and  27. 
10.     16  and  4. 


ANALYSIS  AND  PROPORTION. 

ST^Ir,     III.  Ex.,  I.  If  15  boxes  of  oranges  cost  $G0,  what 

will  17  boxes  cost? 

Opkuation  by  Analysis.  If  lo  boxes  cost  $60,  1  box  will  cost  ^ 

4  of  $60,  and  17  boxes  will  cost  17  X  t\  of 

^60  X  17 ^60.     Cancelling  and  multiplying,  the  re- 

X^        _  $>ow,     n  .  ^^^^  .^  ^^^^ 

Operation  by  Pkopoutiox.  $60,  the  price  of  15  boxes,  must  bear  the 

15  :  17  ==:  ^60  :  Ans.  same  relation  to  the  price  of  17  boxes  that 

4  15  bears  to  17.     We  have  then  three  terms 

17  X  ^00  __  ^gg  ^^^^  of  a  proportion  (15  :  17  =  $60  :  ),  and  can 

1^                      '  find  the  fourth  by  multiplying  the  second 

and  thu'd  together,  and  dividing  the  product  by  the  first.     Hence  wo 

derive  the  following 


4 


250  PROPOUTION. 

Et'LE  FOR  Simple  Proportion.  Mahe  the  mrniher  that  (s 
of  the  same  hind  as  the  required  answer  the  third  term.  If  the 
answer  should  he  greater  than  the  third  term,  make  the  larger  of 
the  other  two  numbers,  upon  which  the  answer  depends,  the  second 
term,  and  the  smaller  the  first.  If  it  should  he  less,  mahe  the 
smaller  number  the  second  term,  and  the  larger  the  first.  Multiply 
the  means  together,  and  divide  their  product  hy  the  first  extreme. 

Nqte.  —  Analysis  is  the  more  natural  and  philosophical  method  of 
solving  arithmetical  questions ;  but  the  principles  of  Proportion  are 
applicable  to  certain  classes  of  examples.  It  is  recommended  to  the 
pupil  to  perform  the  following  examples  in  both  ways.  He  should, 
at  Jeast,  perform  a  sufficient  number  of  them  by  Proportion  to  fix  the 
method  in  his  mind. 

Examples. 

1.  If  2  men  build  17  rods  of  wall  in  a  week,  how  many  rod:* 
will  100  men  build  in  the  same  time  ?  ,>:  o  ^^    ^'  *"'^ 

We  make  17  rods,  which  is  of  the  same  deno;t=iaation  as  the  required 
answer,  the  third  term.  As  100  men  will  buiitl  more  wall  than  2  men, 
we  make  100  the  second  term,  and  2  the  first  term,  and  the  statement 
is,  2  :  100  —  17  :  850.     Ans.  850  rods. 

2.  If  9  lbs.  of  lead  make  150  bullets,  how  many  bullets  can 
be  made  from  105  Ibs^?  Ans.  1,750    bullets. ' 

3.  If  65  pairs  of  boots  can  be  made  from  75  lbs.  of  calfskin, 
how  many  pairs  can  be  made  from  850  lbs.  ?        '  Ans.  73 6§  prs. 

4.  How  many  tons  of  hay  can  be  made  from  750  acres  of  land, 
if  13  tons  can  be  made  from  3  acres  ?  Ans.  3250  tons. 

5.  If   $2000000  will  support  an  army  of  500000  men  a  day,  • 
how  many  men  can  be  kept  for  $400000  ?        Ans.  100,000  men. 

6.  If  $500  purchase  200  hats,  how  many  hats  can  be  pur- 
chased for  $87^^  ?  Ans.  35  hats. 

7.  If  $800  yield  %i)(d  of  interest  in  a  certain  time,  what  will 
$390  yield  at  the  same  rate?  Ans.  $27.30. . 

8.  If  16  horses  eat  a  certain  quantity  of  hay  in  13  weeks,  how 
long  would  the  same  quantity  last  24  horses  ?        Ans.  8f  weeks. 

9.  What  time  would  be  required  for  5  men  to  mow  an  acre  of 
land,  if  2  men  can  mow  it  in  1  J-  days  of  10  hours  in  length  ? 

Aris.  6  hours. 


ANALYSIS  AND  SIMPLE  PIIOPORTIOX.  2ol 

10.  If  o3v)  bushels  of  plaster  were  sufficient  for  the  dressing 
of  3^  acres  of  land,  what  would  be  required  for  17^  acres  ? 

Ans.  2500  bu. 

11.  If  95  acres  of  land  were  mowed  by  20  men  in  2  days  of 
12  hours  each,  how  much  time  would  be  required  for  3  mowing 
machines  to  do  the  same  work  (1  machine  being  equal  to  4: 
men)  ?  Ans,  3  d.  4  h. 

12.  If  a  Graham  loaf  weighs  1  lb.  2  oz,  when  flour  is  worth 
$7 J-  a  bbh,  what  should  it  weigh,  selling  at  the  same  price,  when 
flour  is  $6  per  bbL  ?  Ans.  1  lb.  6 J  oz. 

13',  If  400  lbs,  of  coal  are  required  to  run  an  engine  12  hours, 
what  number  of  tons  will  be  required  to  run  three  similar  en- 
gines for  30  days  constantly  ?        )  1  >-   /: 

14.  If  it  requires  13  days  of  9  hours  each  to  do  a  piece  of 
work,  liow  many  days  of  14  hours  each  will  be  required  to  do 
the  same  work  ?  Ans,  8 y^j. 

15.  If  soda-crackers  can  be  sold  at  10  cents  a  pound  wheu 
flour  is  worth  $6.50  per  bbl,,  for  what  can  they  be  sold  when 
flour  is  worth  $9.75  per  bbL  ?        /  h^  '' 

16.  If  it  requires  30  men  to  do  a  piece  of  work  when  they 
work  1 1  hours  a  day,  what  number  will  be  required  when  they 
work  15  hours  a  day?       >  .C 

17.  If  30  men,  working  11  hours  a  day,  can  do  a  piece  of  work 
in  a  certain  time,  how  many  more  men  must  be  employed,  when 
it  is  half  done,  to  finish  it  in.  the  same  number  of  days,  working 
10  hours  a  day?  Ans.  3  more. 

18.  A  pole  10  ft.  long  casts  a  shadow  of  7  ft.  1  in.  at  8  o'clock ; 
what  is  the  height  of  a  flagstaff  that  casts  a  shadow  58  feet  at  the 
same  time  of  day?  Ans.  81|^  ft. 

19.  If  my  friend  lends  me,  $7000  for  15  days,  for  what  time 
should  I  lend  him  $4500  to  requite  the  favor?  Ans.  23^  d, 

20.  If  my  friend  lend  me  money  for  4  months  when  interest  is 
10  per  cent.,  for  what  time  should  I  lend  him  the  same  sum  when 
interest  is  7  per  cent.  ?      jj*  -^ 

21.  A  sail  vessel  has  10  hodrs  the  start  of  a  steamer,  and  sails 
at  the  rate  of  7  miles  an  hour,  while  the  steamer  sails  16  mile^ 
an  hour ;  when  will  the  steamer  overtake  \lie  sail  vessel  ?  Ans.  7|  h. 


252  PROPORTION. 

22.  A  deer,  150  rods  before  a  hound,  runs  30  rods  a  minute; 
the  hound  follows  at  the  rate  of  42  rods  a  minute  ;  in  what  time 
will  the  deer  be  overtaken  ?  /  %    ',    j  l^^Q  \\    ]   \    (  I  l  ^ 

23.  Two  armies  are  on  opposite  sides  of  a  river,  one  being 
500  miles  east  and  the  other  250  miles  west  of  it,  and  marching 
jowards  each  other,  the  first  at  the  rate  of  15  and  the  other  of  18 
miles  in  a  day ;  in  what  number  of  daj's  will  they  meet,  and 
where?  ^?is.  IGf  d. ;  50  m.  east. 

24.  If  2  lbs.  5  oz.  of  wool  make  1  yd,  of  cloth  32  inches  wide, 
how  much  will  make  a  yard  1^  yards  wide  of  the  same  quality  ? 

Ans.  3  lb.  47jV  oz. 

25.  How  much  cloth  that  is  |  yd.  wide  will  cover  24  tables 
6  ft.  long  and  3  ft.  wide  ?  Ans.  64  yds. 

26.  If  it  requires  10  yards  of  cloth  that  is  1 1  yds.  wide  to  make 
a  garment,  hovy  much  will  be  required  of  that  which  is  1 1  yds, 
wide  ?       / 

27.  How  many  yards  of  cambric  24  inches  wide  will  be  re- 
quired to  line  14^  yards  of  sillc  which  is  22  inches  wide  ?        .  J^ 

28.  How  long  must  a  piece  of  land  be  to  contain  3  acres,  if  it* 
is  4  rods  wide  ? 

29.  If  400  bushels  of  potatoes  were  bought  for  $350.90,  and 
sold  for  $425.50,  what  would  be  gained  on  25  bushels  ? 

30.  A  man  failing  in  trade  owes  $7,865  ;  his  property  is  sold 
for  $4385.70  ;  what  will  he  be  able  to  pay  to  a  creditor  to  whom 
he  owes  $1500  ? 

31.  If  it  costs  $30  to  paint  the  outside  of  a  house  40  ft.  by  30 
■^t.  and  25  ft.  high,  what  will  it  cost  to  paint  one  50  ft.  by  40,  of 
the  same  height?  A7is.  |38f. 

32.  If  a  building  13  ft.  high  casts  a  shadow  of  4  ft.,  what  is 
the  length  of  a  shadow  cast  by  a  tree  346|  ft.  high  at  the  same 
time?  ,     ,      .' 

33.  If  $110  be  paid  for  3  T,  it  cwt.  20  lb.  of  hay,  what  will 
be  the  cost  of  6  T.  3  cwt.  3  lbs,  ?  Ans.  $78.59 ^^g^. 

34.  If  a  hind  wheel,  which  is  8|-  feet  in  pirciimfeTence,  turns 
800  times  in  a  journey,  how  many  times  will  the  fqre  wlicel 
which  is  6^  feet  in  circumference,  turn  in  the  same  journey? 


ANALYSIS  AND   SIMPLE   PROPORTION.  2lB 


85.  The  weight  of  a  cubic  foot  of  water  being  62^  lbs.,  how 
many  pounds'  weight  will  a  tank  contain  which  is  2^  ft.  square  at 
the  bottom,  and  4  ft.  high  ?  /  '    >   2  ^^ 

36.  If  15  A.  2  R.  20  r.  pasture  2  cows  a  certain  time,  what 
will  be  required  to  pasture  25  cows  the  same  time  ?y  ^r ,     v.  / 

37.  How  many  yds.  of  cloth  can  be  bought  for  17£.  o  s.,  if  10  s. 
were  paid  for  5  quarters,  or  1  Ell  English  ? 

38.  What  is  the  value  of  7  cords  3  cord  feet  of  wood  at  the 
rate  of  $18  for  2  cords  5  cord  feet  ? 

39.  If  5|  cwt.  of  leather  pay  for  ^  of  an  acre  of  land,  how 
many  pounds  would  be  required  to  pay  for  1^-  sq.  rods? 

Ans.  12-j3/^  lbs. 

40.  If  9  oz.  10  pwt.  3  gr.  of  gold  be  worth  8150,  what  will 
be  the  value  of  7  lb.  5  oz.  5  pwt. ?  /■  f  ^  1/  o\  /  '^}'iJ 

41.  If  15|.lbs.  of  tea  pay  for  48  lbs.  12  oz.  of  butter,  how  many 
pounds  of  butter  can  be  purchased  with  a  box  of  tea  which  con- 
tains 42 1  lbs.?  j  "  ,.        .   :  '  :■ 

42.  If  .80  Jy  acres  of  land  be  worth  175.20,  what  is  the  value 
^f  .373^  acres  ?  Ans.  $35,053+. 

Note.  —  Perform  the  following  examples  by  analysis. 

43.  If  a  cow  and  a  calf  sell  for  $27,  and  the  value  of  the  calf 
is  I  that  of  the  cow,  what  is  the  value  of  each  ?^ 

44.  The  sum  of  the  ages  of  a  father  and  son  is  48  years,  that 
of  the  son  being  |  of  the  age  of  the  father ;  what  is  the  age  of 
each  ?        Jcr 

45.  By  one  pipe  a  cistern  can  be  emptied  in  2  hours ;  by  an- 
other it  can  be  emptied  in^  3  hours :  in  what  time  can  it  be  emp- 
tied if  both  are  running?  -^^^       f  "^"  '- i"    Z"  YJ/  -  ''       / 

46.  A  certain  cistern  h^s  .^  pipes ;  t«e"fSst  will  4mp£y  it  in  5 
hours,  the  second  in  4  hours,  and  the  third  in  10  hours  ;  in  what 
time  will  all  empty  it  ?      |     j^  . 

47.  A  certain  piece  of  work  can  be  performed  by  A  in  8  days, 
by  B  in  10  days,  and  by  ^  in  1^  days;  in  what  time  can  all  d^^ 
it  working  together  ?    -5    V':  ^. 

48.  In  what  time  can  £  and  B  do  it  together  ?  ^i     i: 


254  PROPORTION. 

49.  In  -what  time  can  A  and  C  do  it  together  ? 

50.  In  what  time  can  B  and  C  do  it  together  ? 

51.  A  cistern  has  one  pipe  which  will  fill  it  in  5 J-  hours,  and 
another  which  will  empty  it  in  33  h. ;  in  what  time  will  it  be 
tilled  with  both  open  ?  Ans.  6f  h. 

52.  If  A  and  B  can  do  a  piece  of  work  in  3  mo.  10  d.,  and  A 
alone  can  do  it  in  5  mo.,  in  what  time  will  B  do  it  alone  ?  Ans.  10  mo. 

53.  If  A  and  B  can  dig  a  trench  in  6|  days,  and  B  can  do  it 
alone  in  10  days,  in  what  time  can  A  do  it  alone  ?    :;  2     '       s 

54.  If  A  and  B  can  do  it  in  5^  days,  and  A,  B,  and  C,jcan 
do  it  in  o^  days,  in  what  time  can  C  do  it  alone  ? 

55.  If  A  and  B  can  do  a  piece  of  work  in  4|  days,  and  A  and 
C  can  do  it  in  5  ^  days,  and  B  and  C  in  6y\  days,  in  what  time? 
will  all  do  the  work  together,  and  in  what  time  will  each  do  it 
alone  ?  Ans,  A,  B,  C,  3^^d. ;  A,  8  d. ;  B,  10  d. ;  C,  i6  d. 

56.  A  man  sold  a  load  of  coal,  containing  |  T.,  at  $.75  for  a 
hundred  lbs.,  and  received  pay  in  corn  at  $.871  per  bushel ;  h<;w 
many  bushels  did  he  receive  ? 

57.  Purchased  a  number  of  pieces  of  goods,  each  contain- 
ing 22  yards,  at  $4  for  3  yards,  and  sold  them  at  $5  for  2 
yards,  and  gained  $154  on  the  lot;  how  many  pieces  were  pur- 
chased ?  Ans.  6  pieces- 

58.  "If  a  third  of  6  be  3,  what  will  a  fourth  of  20  be?" 
^p"  For  Dictation  Exercises,  see  Key. 

ANALYSIS  AND  COMPOUND  PROPORTION. 

375»  III.  Ex.  If  10  gas-burners  consume  800  feet  of  gas 
in  3  hours,  how  many  feet  will  12  burners  consume  in  15  hours? 

Operation  by  Analysis.  If  10  burners    consume  800 

0  ^  feet  of  gas  in  3  hours,  1  burner 

800  X  X^  X  ^0  __  .        ,         will  consume  ^\  of  800  feet,  and 

j0-y-0     -  —  4b00  tt.,  Ans.    ^^  burners  will  consume  12  X 

^  J^  of  800  feet  in  the  same  time ; 

if  this  is  consumed  in  3  hours, 
in  1  hour  there  will  be  consumed  ^  of  12  X  iV  ^^  ^^^  ^^^^'  ^"^  ^  ^^ 
hours,  15  X  i  of  12  X  1^^  of  800  feet. 


ANALYSIS   AND   COMPOUND    rROPOllTION.  255 


Opeuation  by  Pkopoktiox.  800  feet  being  of  the  same  kind 

10  :  12  )  __  gQQ  .  j^^Q^  as  the  answer,  we  make   it  the 

3  :  15  *  third  term.     But  the  number  of 

6         ^  feet  required  depends  upon  two 

800  yC  ^^  X  X^  _   ,Q^^ /.      J         conditions:  1st,  the  number  of 

X0X^  ~  *'         '    gas-burners,  and,  2d,  the  number 

^  of  hours  during  which   they  are 

burning.  We  shall  deal  with  each 
condition  separately.  As  12  burners  will  consume  more  gas  than  10 
burners,  we  make  12  the  second  term  and  10  the  first  term  ;  and  as 
they  will  burn  more  in  15  hours  than  in  3  hours,  we  make  15  the  sec- 
ond term  and  3  the  first  term.  We  find  the  fourth  term,  as  in  simple 
proportion,  hij  dividing  the  product  of  the  means,  12  X  15  X  800,  hy 
the  product  of  the  first  extremes,  10  X  3. 

The  process  may  sometimes  be  shortened  by  cancelling  the  terms 
as  they  stand  in  the  proportion,  remembering  that  the  numbers  which 
constitute  the  first  terms  are  divisors,  and  those  which  constitute  the 
second  and  third  are  multipliers.     Thus, 

2 


'^^i  ^.^1^:800  :4800  ft.,    Arts, 


From  the  above  we  derive  the  following 

Rule  for  performing  Examples  by  Compound  Pro- 
portion. Make  the  number  that  is  of  the  same  kind  as  the  an- 
swer the  third  term.  Take  each  two  numbers  that  are  of  the  same 
kind,  and  consider  whether,  depending  upon  them  alone,  the  answer 
will  be  greater  or  less  than  the  third  term.  Arrange  them  as  in 
simple  proportion.  Divide  the  continued  product  of  the  second 
and  third  terms  by  the  continued  product  of  the  first  terms. 

Examples. 

1.  If  $90  is  paid  for  the  labor  of  20  men  6  days,  what  should 
be  paid  for  5  men  8  days  ?  Ans.  $30. 

2.  If  85  tons  of  coal  were  required  to  run  6  engines  17  hours 
a  day,  what  number  would  be  required  to  run  25  engines  12  hours 
a  day  ?  Ans.  250  tons. 

3.  If  120  rods  of  watl  were  laid  by  72  men  in  33  days  of  14 


256  PROPORTION. 

hours  each,  how  many  rods  would  be  laid  by  88  men,  working  12 
hours  a  day  for  3^  days  ?  Ans.  13^  rds. 

4.  If  the  wages  of  75  men  for  84  days  were  $G8.75,  for  how 
many  days  could  90  men  be  employed  for  $41.25  ?        Aiis.  42  d. 

5.  If  the  freight  on  450  lbs.  of  merchandise  is  30  cents  for  26 
miles,  how  many  miles  can  3  tons  be  carried  for  $4?  Ans.  26  m. 

6.  If,  when  flour  is  $7.50  per  bbl.,  a  3-cent  loaf  weighs  2  oz., 
what  should  a  12-cent  loaf  weigh  when  flour  is  |16?  Ans.  3|  oz. 

7.  If  ^  loaf,  which  sells  for  10  cents  when  wheat  is  $2  a  bushel, 
weighs  1^  lbs.,  what  is  the  price  of  wheat  when  a  6-cent  loaf 
weighs  1|  lbs.?  Ans.  $1.44. 

8.  If  500  lbs.  of  wool  worth  42  cents  a  lb.  is  given  for  75 
yds.  of  cloth  1|  yds.  wide,  how  much  wool  worth  36  cents  a  lb. 
should  be  given  for  27  yds.  that  is  1^  yds.  wide?    Ans.  193}^  lb. 

9.  If  it  costs  $135.02  to  carry  7  cwt.  2  qr.  15  lb.  a  distance  of 
64  miles,  what  will  be  the  cost  of  carrying  1 1  cwt.  2  qr.  a  dis- 
tance of  15J-  miles  ?  Ans.  $49,157+. 

10.  If  25  men,  in  9^  days  of  10  hours  each,  build  200  rods 
of  wall,  how  many  rods  would  be  built  in  1  day  of  12  hours  by 
1^  men?  Aiis.  12 if  rds. 

11.  If  11  men,  in  20  days  of  12  hours  each,  can  build  a  wall 
48  feet  long,  8  feet  high,  and  3  feet  thick,  in  how  many  days  can 
15  men,  working  10  hours  a  day,  build  440  ft.  of  wall  12  ft.  high, 
and  4  ft.  thick  ?  Ans.  322|  d. 

12.  How  many  men  will  be  required,  working  12  hours  a  day 
for  250  days,  to  dig  a  ditch  750  ft.  long,  4  ft.  wide,  and  3  ft.  deep, 
if  it  requires  27  men,  working  13  hours  a  day  for  62  days,  to  dig 
a  ditch  403  ft.  long,  3  ft.  wide,  and  3  ft.  deep?  Ans.  18  men, 

13.  If  5  men,  working  12  hours  a  day  for  8  days,  cut  44  loadi 
of  wood,  each  8  ft.  long,  4  ft.  wide,  and  4  ft.  6  in.  high,  in  how 
many  hours  would  16  men  cut  49^  loads  8  ft.  long,  4  ft.  wide, 
and  5  ft.  6  in.  high?  A?is.  41i  hours. 

14.  If  7  cases  of  boots  can  be  made  by  9  men  laborino-  12 
hours  a  day  for  7  days,  what  length  of  time  will  be  required  for 
3  men  and  4  boys  (2  boys  being  equal  to  a  man  and  a  half), 
laboring  11  hours  a  day,  to  make  33  cases?  Ans.  54  d. 


ANALYSIS  AND  COMPOUND  PROPORTION.  257 

15r  If  421 1-  rods  of  fence  which  is  4  feet  high  be  built  by  9 
men  in  10  days  of  9  hours  each,  how  many  rods  of  a  fence  5  ft. 
high  will  be  built  by  8  men  in  6  days  of  8  hours  each  ? 

16*  What  length  of  canal  27  ft.  wide,  24  ft.  deep,  can  be  dug 
by  250  men  in  100  days  of  12  hours  each,  if  750  men  in  3 
months  of  25  days  each,  working  11  hours  a  day,  could  dig  4 
miles  of  a  canal  30  ft.  wide  and  10  ft.  deep  ? 

Arts.  7  fur.,  7  rods,  5  ft.,  2|  in. 

17!"  If  the  type  for  a  book  of  84  pages,  50  lines  to  a  page, 
lines  averaging  8  words,  1^-  syllables  to  a  word,  2^  letters  to  a 
syllable,  was  set  by  2  men  in  5  days  of  12  hours  each,  how  many 
pages  of  a  book,  each  page  containing  75  lines,  averaging  5^ 
words  each,  2  syllables  to  a  word,  3  letters  to  a  syllable,  would 
be  set  by  7  men,  laboring  8  hours  a  day  for  the  working  days  of 
a  week?  Ans,  142y\  pages. 

18^  If  12  men,  in  2  months  of  4  weeks  each,  working  6  days 
per  week,  12  hours  a  day,  can  set  the  type  for  12  books,  of  600 
pages  each,  120  lines  to  a  page,  20  words  to  a  line,  10  letters  to 
a  word,  in  how  many  months  of  4^  weeks  each  will  7  boys, 
working  4  days  per  week,  16  hours  a  day,  set  the  type  for  6 
books,  of  500  pages  each,  150  lines  to  a  page,  lines  averaging  24 
words,  4^  letters  to  a  word,  each  boy  doing  ^  of  the  work  of  a 
man?  Ans.  l^f^jj  months, 

19t  If  it  requires  15  yds.  of  silk,  f  yd.  wide,  to  line  a  cloak 
made  of  12.25  yds.  of  cloth,  1  yd.  1-i  qrs.  wide,  how  many  yards 
of  silk,  I  yd.  wide,  will  be  required  to  line  a  cloak  made  of  8.33  J 
yds.,  4^  qrs.  wide?  Ans.  4jYf  J^s. 

20r  If  4  men  dig  a  cellar  33.75  ft.  long,  18  ft.  wide,  and  9.6  ft. 
deep,  in  4.5  days  of  11.25  hours  each,  in  how  many  days  of  11.7 
hours  will  15  men  dig  a  cellar  15.2  yds.  long,  7.8  yds.  wide,  and 
10.8  ft.  deep,  the  latter  cellar  being  twice  as  hard  to  dig  as  the 
former  ? 

^^  For  Dictation  Exercises,  see  Key. 

17 


258  PROPORTION. 

PARTNERSHIP,   OR  PARTITIVE  PROPORTION. 

S'^0.    Partnership  is  the  association  of  two  or  more  persons 

for  the  transaction  of  business.     The  persons  thus  associated  are 

called  Partners. 

The  profits  and  losses  are  generally  shared  by  the  partners  in 

proportion  to  the  stock  each  has  in  trade,  and  the  time  it  is 

employed. 

III.  Ex.     A  and  B  formed  a  partnership  for  one  year.     A  put 

in  ^800,  and  B  $700.     They  gain  $375.     What  is  the  share  of 

each? 

Operation.  If  ,91500  gain  $375,  $1  will 

$800  +  $700  =  $1500,  whole  stock.  j^  _l^  ^f  ^375  ^^^  ^8(^0 
^Vo  of  $375 :=  $200,  A's  share.  .^-n  g^l^sOO  X  -,i^^  of  $375 
iVA  of  $375  z=z  $175,  B's  share.         ^  ^200,  A's  share,  and  $700 

will  gain  700  X  i^  of  $375  =  $175,  B's  share.     Hence  the 

KuLE  FOR  Simple  Partnership.  Apportion  the  gain  or 
loss  among  the  partners  according  to  their  stock  in  trade. 

Examples. 

1.  A  and  B  form  a  partnership,  A  putting  in  $4000,  and  b 
$3000  ;  they  gain  $3000  ;  what  is  the  share  of  each  ? 

Arts.  A,  $1714f  ;  B,  $1285^ 

2.  E,  S,  and  Z  invest  in  trade  as  follows :  E  $500,  S  $800,  and 
Z  $300  ;  they  gain  $375.75  ;  what  is  the  share  of  each  ? 

Ans  £,$117.42^3^;  S,  $187.87i  ;  Z,  $70.45-j5g. 

3.  Two  traders,  A  and  B,  shipped  coal  from  Philadelphia  to 
Boston.  A  had  on  board  350  tons,  and  B  900  tons.  It  became 
necessary  to  throw  overboard  250  tons.  What  was  the  loss  of 
each?  Ans.  A,  70  tons;  B,  180  tons. 

4.  Four  persons,  H,  I,  K,  and  L,  engaged  in  an  adventure 
H  furnished  $275  ;  I,  $640 ;  K,  $330  ;  L,  as  much  as  II  and  K. 
They  gained  $4675.     What  was  the  share  of  each  ? 

Ans.  H,  $694.93^9^;  I,  $1617.29§f;  K,  $833.91ff ;  L, 
^1528.85^^7. 

5.  A  bankrupt  owes  to  Q  $800,  to  R  $350,  and  to  X  and  Y 
$500  each;    his  whole  property   sells  for  $1584.80,  of  which 


SIMPLE  PARTNERSHIP.  ^59 

$158.48  arc  required  to  pay  the  expenses  of  the  sale ;  what  will 
each  person  receive  ? 

Ans,  Q  receives  $530.72|f ;  E,  $232.19/^;  X  and  Y,  each, 
$331.70^^. 

6.  Gerry  &  Frost  enter  into  a  dry  goods  business.  G  puts  in 
250  pieces  of  cotton,  30^  yds.  each,  at  10  cts. ;  100  yds.  broad, 
cloth,  at  $3.25  ;  and  $918.75  in  cash.  F  puts  in  40  pieces  linen, 
12  yds.  each,  at  28  cts. ;  other  goods  to  the  amount  of  |500,  and 
cash  $5G5.60.  On  closing  business,  they  find  they  have  lost 
40  %  of  their  investment.     What  is  the  loss  of  each  ? 

7.  Four  persons,  M,  N,  O,  and  P,  engaged  in  partnership.  M 
put  in  700  bu.  wheat,  at  $1.25;  N,  1000  lbs.  wool,  at  |.37^;  O, 
500  bbls.  flour,  at  $6.50;  P,  2000  bu.  corn,  at  $.90.  Their 
whole  stock  being  destroyed  by  fire,  the  firm  received  insurance 
on  the  goods,  $4200 ;  what  was  the  loss  to  each  partner  ? 

/-.8.  Suppose  the  store  occupied  by  the  above  persons,  owned  by 
A,  B,  and  C,  and  valued  at  $80000,  to  have  been  insured  for  f 
of  its  value.  What  would  be  the  loss  to  each  partner  by  the  fire, 
the  store  being  entirely  consumed,  A  owning  ^,  B  ^,  and  C  the 
remainder?  Ans.  A,  $5000;  B,  $6666f ;  C,  $83331. 

9.  Banks  &  Ward  traded  in  company,  and  gained  $975.  It 
was  agreed  that  B  should  have  $8  of  the  gain  as  often  as  W  had 
$7.     What  was  the  share  of  each?         Ans.  B,  $520  ;  W,  $455. 

10.  Divide  $1500  among  three  persons  so  that  their  shares 
shall  be  in  the  proportion  of  3,  4,  and  5. 

11.  Five  persons.  A,  B,  &c.,  hired  a  pasture  for  $275.  A  put 
in  250  sheep,  B  300  sheep,  C  200  sheep,  D  15  yoke  of  oxen,  and 
E  10  yoke.  One  ox  being  equal  to  10  sheep,  what  was  due 
from  each?        Ans.  A,  $55;  B,  $66;  C,  $44;  D,  $66;  E,  $44. 

12!"  F,  G,  H,  and  I  are  in  partnership  as  stock  brokers.  F 
furnishes  $2000,  G  $2500,  H  $1500,  and  I  75  shares  of  railroad 
stock.  They  gain  $3300,  of  which  I  receives  $1100  ;  what  was 
his  stock  per  share,  and  what  is  the  gain  of  each  of  the  other 
partners?    Jws.  F,  $733^ ;  G,  S916§;  H,  $550;  I,  $  40  a  share. 

13!  Stewart  &  Mills  traded  in  company  for  one  year.  Their 
gain  was  equal  to  20%  of  their  stock.     S's  share  of  the  gain  was 


2  GO  PROPORTION. 

$150,  which  was  |  of  the  whole  gain;  what  was  M's  gain,  and 
what  the  sum  each  invested  ?  V  /^        / 

14?  Adams  &  Brown  built  a  schooner.  A.  furnished  18000, 
and  B.  $1700  and  15000  ft.  of  lumber.  Her  freights  for  the  first 
year  were  $1125,  of  which  B.'s  share  was  $225  ;  what  was  the 
price  of  his  lumber  per  thousand  feet  ?  Ans.  $20  per  M. 

15t  Jones,  Styles  &  Carpenter  enter  into  partnership.  J.  puts 
in  $750,  S.  $420,  and  C.  60  tons  of  coal.  They  gain  $624,  of 
which  C.  is  to  have  J  for  conducting  the  business,  the  balance  to 
be  shared  among  the  partners  in  proportion  to  their  stock  in 
trade.  C.  receives  $390 ;  what  is  his  coal  per  ton,  and  what  are 
the  shares  of  the  other  partners  ? 

S^^  For  Dictation  Exercises,  see  Key.  I' 


COMPOUND  PARTNERSHIP. 

STT.  When  stock  in  trade  is  employed  for  different  periods 
of  time,  the  partnership  is  called  Compound  Partnership. 

III.  Ex.  Three  persons  formed  a  partnership.  A  put  in 
$170  for  9  mo.,  B  $130  for  12  mo.,  and  C  $150  for  8  mo.  They 
gained  $286  ;  what  was  the  share  of  each? 

Operation.  The  use  of  $  1 70 

170  X    9  =  1530  l|ff  X  $286  =  $102,  A's  share,  for  9  mo.z=the  use 
130X  12=lo60  ^ff  X  $286  =  $104,  B's  share,  of  $1530  for  1  mo.; 
150  X    8  =  1200  ii^  X  $286  =   $80,  C's  share,  the  use  of  $130  for 
4290  12  mo. =$1560  for 

1  mo. ;  the  use  of  $150  for  8  mo.  =:  $1200  for  1  mo.  The  amount  in 
trade  Avas,  therefore,  equal  to  $1530  +  $1560  +  $1200,=  $4290,  for 
1  mo. ;  hence  the  gains  should  be  as  follows :  A's,  i||g-  of  $286  rrr 
$102;  B's,l||^of$286  =  $104;  C's,  iff «- of  $286  =$80.    Hence  the 

BuLE  FOR  Compound  Partnership.  Multiply  each  part- 
ner's stock  hy  the  time  it  is  in  trade,  and  apportion  the  gain  or 
loss  according  to  the  products. 


COMPOUND   PARTNERSHIP.  261 

Examples. 
1.  A  and  B  engaged  in  business,  and  gained  $2008.25.     A 
put  in  $4500  for  9  months,  and  B  $5690  for  7  months.     What 
was  the  gain  of  each  ?  Am.  A,  $1012.50  ;  B,  $996.75. 

^  2.  A,  B,  C,  and  D  work  a  mine  in  company.  A  furnishes 
$1400  for  3  years,  B  $500  for  5  years,  C  $1800  for  2  years,  and 
D  $700  for  4  years.  At  the  end  of  5  years  they  divide  $2620 
of  profits  ;  what  is  the  share  of  each  ? 

Ans.  A,  $840  ;  B,  $500 ;  C,  $720  ;  D,  $560. 

3.  Webb,  Clapp,  and  Calhoun  form  a  partnership.  Webb 
puts  in  $8500  for  7  months,  Clapp  $10000  for  4  months,  and 
Calhoun  $6750  for  9  months.  They  lose  $2499.90.  What  is  each 
partner's  loss  ? 

^m.  Webb,  $928.20;    Clapp,  $624;     Calhoun,  $947.70. 

4.  Hooker,  Brown,  and  Lear  traded  in  company.     H.  put  i 
$2500  for  10  months,  B.  $2300  for  11  months,  and  Lear  con- 
ducts the  business,  which  is  considered  equal  to  $2000  in  trade, 
for  12  months.     They  gain  $1486.     What  should  each  receive  ? 

Ans.  Hooker,  $500;  Brown,  $506;  Lear,  $480. 

5.  Four  persons,  J,  K,  L,  and  M,  loaned  money  as  follows : 
J  $1500  for  5  years,  K  $750  for  3  years,  L  $1700  for  2^  years, 
and  M  $950  for  4  years.  They  received  of  interest  money  $1246. 
Wliat  was  the  share  of  each,  and  what  the  rate  per  cent.  ? 

Ans.  J,  $525  ;  K,  $157^;  L,  $297^;  M,  $266;  rate,  7%. 

6.  A,  B,  and  C  formed  a  copartnership.  A  furnished  |-  of  the 
capital  for  6  months,  B  ^  of  the  capital  for  10  months,  and  C  the 
balance  for  12  months.  The  whole  gain  was  $1560.  What  was 
the  share  of  each  ?  Ans.  A,  $480  ;  B,  $600 ;  C,  $480. 

■^  7.  Hooker  &  Brown  were  in  business  together  for  3  years,  and 
gained  $5750.  Hooker  put  in  $2000  for  the  first  year,  and  $1500 
for  the  other  two ;  Brown  put  in  $2500  for  the  first  two  years, 
and  $1500  for  the  last  year.     What  was  the  gain  of  each  ? 

Ans.  Hooker,  $2500  ;  Brown,  $3250. 

8.  A  and  B  received  $857.50  for  grading  a  road.     A  furnished 

5  hands  for  20  days,  and  6  others  for  15  days;  B  furnished  10 

hands  for  12  days,  and  9  others  for  20  days.     What  was  the 

share  of  each  contractor  ?  Ans.  A,  $332.50 ;  B,  $525. 


2G2  PROPORTION. 

9.  Lincoln  and  Hurd  hired  a  pasture,  for  which  they  paid  8117. 
Lincoln  jut  in  217  head  of  cattle  for  20  days,  150  for  5  days, 
189  for  10  days,  and  500  for  7  days;  Hurd  put  in  650  head  for 
6  days,  48  for  15  days,  and  400  for  11  days.  "What  should  each 
pay  ?  Ans.  Lincoln,  $62.88  ;  Hurd,  $54.12. 

4  10.  Jones  and  Avery  engaged  in  business  as  brokers  for  the 
year  1862.  Jan.  1,  Jones  advanced  $8600  and  Avery  $1250; 
April  1,  Tyler  was  admitted  to  the  firm  whh  |1500;  June  1, 
Childs  was  admitted  with  $1200;  Sept.  1,  Hewins  with  $1800; 
and,  Nov.  1,  Jenkins  with  $2550.  The  losses  for  the  year  were 
^7560  ;  what  was  the  loss  of  each  partner  ? 

Jws.  Jones,  $3534y6j.;  Avery,  $1227f\  ;  Tyler,  $1104f-j  ; 
Childs,  $687y\  ;  Hewins,  $589yiy;  Jenkins,  $417y^^. 
"n  lit  Wallis  and  Winn  engaged  in  trade.  The  former  had  in 
$900  from  January  1  till  April  1,  when  he  withdrew  $450  ; 
July  1  he  added  $600.  The  latter  had  in  $2000  from  Feb.  1  to 
Oct.  1,  when  he  added  $200  ;  Nov.  1  he  withdrew  $800.  The 
whole  gain  during  the  year  was  $2500  ;  what  was  the  share  of 
each.  Ans,  Wallis,  $825/ij59  ;  Winn,  $1674^3|. 

wl2f  Ames  &  Rice  ran  a  steamer  for  3  years.  Ames  furnished 
t|3000  for  the  first  10  months,  when  he  added  $1000  more,  and 
at  the  end  of  the  second  year  $500  more.  Rice  put  in  $2500 
for  the  first  18  months,  when  he  put  in  $3500  more.  At  the  end 
of  the  third  year  they  found  their  loss  to  be  $5565  ;  what  should 
each  sustain  ? 

13!  D,  E,  and  F  hired  a  pasture  on  the  20th  of  May  for  5 
months,  paying  $125  for  its  use.  On  that  day  D  put  in  200 
sheep,  E  150,  and  F  80;  June  20,  D  put  in  40,  E  200,  and  F 
275  ;  July  20,  D  took  out  100,  E  75,  and  F  put  in  80  ;  Sept.  20,  D 
put  in  25,  and  E  and  F  took  out  200  each.    What  should  each  pay  ? 

14.  Weeks,  Wyman  &  Wentworth  engaged  in  business  for  1 
year.  Jan.  1,  each  put  in  $4000 ;  March  1,  Weeks  and  Wyman 
put  in  $1500  each,  and  Wentworth  withdrew  $600  ;  Aug.  1, 
Weeks  put  in  $800,  Wyman  withdrew  $300,  and  Wentworth 
put  in  $1000;  Oct.  1,  Weeks  withdrew  $400;  Nov.  1,  Wyman 
put  in  $650,  and  Wentworth  put  in  $1500.  At  the  end  of  the 
year  they  divided  $3500  profits.     What  was  the  gain  of  each  ? 


REVIEW.  263 

\ 

ISr  A,  B,  C,  and  D  put  $5700  in  trade.     A's  money  was  in 

8  months,  and  his  gain  was  |160  ;  B's  was  in  5  months,  and  his 

gain  was  $200;  C's  was  in  2  months,  and  his  gain  was  $18); 

D's  was  in  6  months,  and  his  gain  was  $240.     What  stock  did 

each  have  in?     A?is,  A,  |600  ;  B,  $1200 ;  C,  $2700;  D,  $1200. 

1^  For  Dictation  Exercises,  see  Key.    ^'- 

S78.     Questions   for   Review. 

Ratio.  —  AVhat  is  ratio  ?  what  is  arithmetical  ratio  ?  geometrical 
ratio  ?  What  is  the  first  term  of  a  ratio  called  ?  the  second  term  ? 
hoth  terms  when  taken  together  ?  What  is  a  ratio  of  equality  ?  of 
greater  inequality?  of  less  inequality?  Give  an  example.  In  what 
respect  do  ratios  resemble  fractions  ?  How,  then,  may  ratios,  at  any 
time,  be  written  ?  How  do  you  multiply  a  ratio  ?  how  divide  a  ratio  ? 
Suppose  you  multiply  or  divide  both  terms  by  the  same  number  ? 
What  is  a  simple  ratio  ?  a  complex  ?  a  compound  ratio  ?  How  do  you 
reduce  a  complex  ratio  to  a  simple  one  ?  a  compound  ratio  ?  Write  a 
simple  ratio ;  a  complex  ratio  ;  a  compound  ratio. 

PiiOPORTiON.  —  What  is  proportion  ?  Explain  the  proportion  2  :  4 
=  7  :  14.  What  are  the  1st  and  4th  terms  called  ?  the  2d  and  3d  ? 
the  1st  and  3dP  the  2d  and  4th?  the  1st  and  2d?  What  is  inverse 
proportion?  compound  proportion?  AVhat  is  a  mean  proportional 
between  two  numbers  ?  Upon  what  important  principle  does  the  solv-. 
ing  of  examples  by  proportion  depend  ?  Write  a  proportion,  and 
illustrate  that  principle.  How  can  you  find  an  extreme,  when  the  othet 
three  terms  are  given  ?  how  a  mean  ?  how  a  mean  proportional  be- 
tween two  given  numbers  ?  Give  the  rule  for  solving  an  example  by 
simple  proportion,  and  illustrate  it  by  an  example  of  your  own.  Per- 
form the  same  example  by  analysis.  What  is  meant  by  analysis  ?  Give 
your  rule  for  solving  an  example  by  compound  proportion,  and  illus. 
trate  it.  In  solving  any  example  by  proportion,  the  two  terms  of  t\ 
ratio  must  be  of  the  same  kind;  why? 

Partnership.  —  What  is  partnership  ?  Who  are  the  partners  ? 
How  are  ])rofits  and  losses  usually  shared  ?  What  is  simple  partner- 
ship ?  {Alls.  It  is  partnership  where  persons  enter  into  business  for 
the  same  time.)  How  do  you  find  each  person's  share  of  gain  or  loss 
in  simple  partnership  ?  AVhat  is  compound  partnership  ?  How  do 
you  find  the  shares  of  gain  or  loss  in  compound  partnership  ?  Why  is 
Vartnership  sometimes  called  partitive  proportion  ? 


264  INVOLUTION. 


yiNVOLUTION. 

379.  Involution  consists  in  raising  a  number  to  a  required 
power.     (Art.  89.) 

380.  The  required  power  is  indicated  by  a  small  figure, 
called  the  index  or  exponerit,  placed  at  the  right,  and  a  little 
above  the  number.     (Art.  90.) 

38  !•  The  Jirst  power  of  a  number  is  the  number  itself.  The 
second  power  or  square  of  a  number  is  obtained  by  using  the 
number  as  a  factor  twice.  The  third  power  or  the  cube  results 
from  using  the  number  as  a  factor  three  times,  and  so  on. 

Note.  —  The  most  important  applications  of  Involution  are  in  the  use 
of  the  second  and  third  powers. 

38S*    Any  power  may  be  obtained  by  the  following 

Rule.  Einploy  the  given  number  as  a  factor  as  many  times 
as  there  are  units  in  the  exponent  of  the  required  power. 

Examples. 

1.  Find  the  squares  of  the  integers  from  1  to  25  inclusive, 
Qnd  commit  them  to  memory.* 

/Numbers,  1,  2,  3,  4,  5,  6,  7,  8, 

Squares,        1,       4,       9,      16,     25,     36,     49,     64, 

I  Numbers,         9,         10,         11,         12,         13,         14,         15,         16, 

•^^^-^  Squares,       81,    100,  121,  144,  169,  196,  225,  256, 

Numbers,      17,         18,       19,         20,        21,         22,         23,        24,        25. 

Squares,    289,  324,  361,  400,  441,  484,  529,  576,  625. 

2.  Find  the  cubes  of  the  integers  from  1  to  10  inclusive,  and 
commit  them  to  memory.* 

C  Numbers,    1,     2,      3,       4,         5,         6,  7,  8,         9,  10. 

-^^*-)  Cubes,       1,  8,  27,  64,  125,  216,  343,  512,  729,  1000. 

*At  the  option  of  the  teacher. 


EVOLUTION. 


265 


S.  922  _  ? 

4.  a)^  =  ? 

5,  .32  —  ? 

6.  (7|-)^  =  ? 

'7.   3.082  zr:  ? 

8.  .3712  _  ? 

9.  43722  zzr? 
^  10.  5.82  =  ? 
ai.  47.62—? 

12.  Q^)2  =.  ? 


^ws.  S464. 
^ns.  If. 
Ans.  .09. 


>4* 

Ans.  9.4864. 
^ws.  .137641. 


13.  (124f)2  =  ? 

14.  97^  =  ? 

15.  5.753  =  ? 

16.  (3|)3z=:? 

17.  IF  =  ? 

18.  1010  —  ? 

19.  (4)12  =  ? 

20.  .59  zn  ? 

21.  Involve  1.3  to  the  6tli  power. 

22.  Raise  18f  to  the  5th  power. 


23.  What  is  the  difference  between  the  square  and  the  cube 


I  CuU  % 


of  24,  ,Tr"'-  ■    ^"- 1  ^   "^'"^  }  0  '^^l  Cu^^-  -  '    ^  -i  "^ 

24.  What  is  the  compound  interest  of  $1.10  for  4  years,  at 
10  per  cent  ?       J^\    J  '':\  ^  j 

25.  How  many  paving  stones  13  inches  square  will  be  required 
to  pave  100  rods  of  a  street  3  rods  in  width.'' 

26.  How  many  dice  measuring  ^  an  inch  each  way  may  be 
made  from  a  cubical  foot  iqf  ivory,  allowing  -j^^  for  waste  in  the 
manufacture?    j    'i|i/  1^ 


EVOLUTION. 

983*    Evolution  *consists  in  finding  the  roots  of  numbers. 

384*  The  root  of  a  number  is  one  of  the  equal  factors  which 
produce  that  number. 

The  square  root  is  one  of  the  two  equal  factors  ^  the  cube  root, 
one  of  the  three  equal  factors ;  the  fourth  root,  one  of  the  four 
equal  factors,  and  so  on. 

385.  a/  is  the  radical  sign,  and  by  itself  signifies  the  square 
root,  and  with  a  figure  above  it,  signifies  the  degree  of  the  root 
indicated  by  the  figure  ;  thus,  ^^27  signifies  the  third  root  of  27. 

The  root  may  also  be  indicated  by  a  fractional  exponent ;  thus, 
16^  (r«ad,  16  to  the  i  power)  =  ^^  16  =:  2. 


5f66  EVOLUTION. 


SQUARE    ROOT. 


38G*  Table,  showing  the  places  occupied  by  the  square  of 
any  number  of  units,  tens  or  hundreds. 

Koots.  Squares. 

1  squared  =:  j 

9       "  =  81 

10     "  =  ioo 

99       "  =  980i 

100      «  =         ioooo 

999       "  =         998001 

38 T,  By  the  above  we  perceive  that  the  square  root  of  any 
whole  number  expressed  by  one  or  two  figures,  must  be  units  ; 
expressed  by  three  or  four  figures,  must  be  units  and  tens  ;  ex- 
pressed by  five  or  six  figures,  must  be  units,  tens,  and  hundreds. 
Plence,  generally,  if  a  number  he  separated  into  periods  of  two 
figures  each,  beginning  with  the  units,  the  number  of  figures  in  the 
square  root  will  be  iridicated  by  the  number  of  periods. 

Note  I.  —  The  left  hand  period  may  contain  but  one  figure. 
Note  II.  —  The  principle  above  elucidated  applies  also  to  decimal  frac- 
tions ;  but  every  period  in  decimal  fractions  must  contain  two  figures. 

388.  That  the  pupil  may  comprehend  the  method  of  ex- 
tracting the  square  root  of  a  number,  we  will  multiply  64  by 
itself,  I.  e.,  square  it,  and  keep  the  separate  products,  instead  of 
reducing  them  and  adding  as  in  ordinary  multiplication. 

G4  X  64  =  (GO  +  4)  X  (60  +  4). 
(1.)  60  X  60  =:  602  —  3^00. 

(2.)      {^4^60}  =2  X  (60   X  4)zn    480. 

(3.)  4X4=  42=      16. 

By  the  above  it  will  appear  that  a  square  whose  root  is  com- 
posed of  tens  and  units,  contains 
(1.)      The  square  of  the  tens  ; 
(2.)      Twice  the  tens  multiplied  by  the  units  ;  and 
(3.)      TRe  square  of  the  units. 


—  4096. 


SQUARE  ROOT. 


267 


Operation. 


Trial  divisor,  2  X  6(0)  z= 

:12(0)\    496 
4  7   496 

True  divisor,  .... 

124       000 

Or  simply i 
4096  (  64 
36 

124  )  496 
496 

389.    We  will  now  extract  the  square  root  of  4096. 

»  i         Separating  the  number 
HP     into  periods  of  two  figures 
4096  (  64     each,  we  find  that  the  root 
6(0)2 ;—  36  ^j2  consist  of  two  figures, 

and  the  square  of  the  tens 
must  be  contained  in  the 
40  (hundreds) ;  the  largest 
square  contained  in  40 
(hundreds)  is  36  (hun- 
dreds), the  root  of  which 
is  6  (tens)  ;  this  we  write 
as  the  tens'  figure  of  the 
root,  and  subtract  its 
square  36  (hundreds)  iGrom  the  40  (hundreds),  and  to  the  remainder  4 
(hundreds),  bring  down  the  next  period,  96. 

This  remainder  (496)  must  contain  two  times  the  tens  of  the  root 
multiplied  by  the  units,  pltcs  the  square  of  the  units,  or  the  jyroduct  of 
two  times  the  tens,  plus  the  units,  multiplied  by  the  units.  If  it  contained 
only  two  times  the  tens  multiplied  by  the  units,  we  should  obtain  the 
units'  figure  by  dividing  the  remainder  (496)  by  two  times  the  tens. 
"We  make  2X6  (tens)  =  12  (tens)  the  trial  divisor,  which  is  contained 
in  49  (tens)  4  times.  AVe  write  4  as  the  units'  figure  of  the  root,  and 
also  at  the  right  of  the  12  (tens),  and  have  124  for  the  true  divisor.  This 
we  multiply  by  4,  and  thus  complete  the  square,  —  obtaining  at  once, 
twice  the  product  of  the  tens  by  the  units,  and  the  square  of  the  units. 
If  the  root  consists  of  more  than  two  figures,  having  obtained  tha 
ftrst  two,  we  can  consider  them  as  tens  in  reference  to  the  next  figure, 
and  proceed  with  them  in  all  respects  as  above.     Thus,  suppose  it  be 


Operation. 
4i38.94  (  64.33-f 
36 


124  )  538 
496 


1283  )  4294 
3849 


12863  )  44500 

38589 

5911 


required  to  extract  the 
square  root  of  4138.94: 
find  the  first  two  places 
as  before  ;  bring  down 
the  next  period,  94,  and 
form  a  new  trial  divisor 
by  doubling  64  (the  root 
already  found)  ;  find  how 
many  times  this,  consid- 
ered as  tens,  is  contained 
in  429  tens,  for  the  third 


268  EVOLUTION. 

figure  of  the  root.     To  obtain  a  fourth  figure  in  the  root;  form  another 
IDeriod  by  annexing  two   zeros,    double  643,  and  so  continue. 

From  the  above,  we  deduce  the  following 

Rule  for  extracting  the  Square  Root  of  a  Num- 
ber. Point  off  the  given  number  into  periods  of  two  figures  each, 
by  placing  a  dot  over  the  units'  Jigure  and  every  alternate  figure 
to  the  left  in  whole  numbers,  and  to  the  right  in  decimals. 

Find  the  greatest  square  number  in  the  left  hand  period,  and 
write  its  root  as  the  first  term  in  the  answer.  Subtract  the  square 
number  from  the  left  hand  period,  and  to  the  remainder  hring 
down  the  next  period  for  a  dividend. 

Take  twice  the  root  already  found  for  a  trial  divisor  ;  rejecting 
the  right  hand  figure  of  the  dividend,  divide  it  by  the  trial  divisor ; 
place  the  result,  as  the  second  term  in  the  root,  also  at  the  right  of 
the  trial  divisor,  mahing  a  true  divisor  ;  multiply  the  true  divisor 
thus  obtained  by  the  last  term  of  the  root,  and  subtract  this  product 
from  the  dividend ;  to  the  remainder  bring  down  the  next  period 
for  a  new  dividend. 

Double  the  terms  of  the  root  already  found  for  a  new  trial 
divisor,  and  proceed  as  before. 

Note  I.  —  When  a  zero  occurs  m  the  root,  annex  a  zero  to  the  trial 
divisor;  bring  down  another  period,  and  proceed  as  before. 

Note  II.  —  If  a  root  figure  proves  too  large,  substitute  a  smaller,  and 
repeat  the  work. 

Note  III.  —  When  a  remainder  occurs  after  all  the  periods  are  brought 
down,  the  root  may  be  more  nearly  approximated  by  annexing  periods  of  • 
zeros,  and  continuing  the  operation. 

Note  IV.  —  The  square  root  of  a  common  fraction  maybe  obtained  by 
■  extracting  the  root  of  both  terms  when  they  are  perfect  squares  ;  when 
they  are  not,  the  fraction  may  first  be  reduced  to  a  decimal. 

Note  V.  — Mixed  numbers  may  be  reduced  to  the  decimal  form,  or  to 
improper  fractions  when  the  denominator  of  the  fractional  part  is  a  squar« 
number. 


SQUARE  ROOT. 


269 


R 

Fig.  1, 

c 

3600  sq.  ft. 

s 

IS 

60  ft. 

D 

Fig.  2. 


390.  The  above  rule  may  be  illustrated  by  diagrams. 

Let  A  B  C  D  (Fig.  1)  represent  a  square 
court  containing  4096  square  feet,  the  length 
of  whose  side  we  wish  to  determine.  Having 
found  (Art.  389)  that  the  greatest  square  of  tens 
in  4096  is  3600,  the  root  of  which  is  6  tens,  we 
deduct  3600  from  4096,  and  have  left  496  square 
feet,  which  are  to  be  disposed  on  two  sides  of 
the  square  already  found.  The  width  of  these 
additions  we  wish  to  ascertain. 

By  extending  the  lines  a  and  b,  we  shall 
divide  the  addition  into  three  parts,  M,  N,  and 
O  ;  M  and  N  having  for  one  side  the  tens  of 
the  root,  and  O  being  a  square  whose  side  is 
equal  to  the  width  of  the  side  additions. 

If  the  496  square  feet  equalled  the  feet  in  the 
side  additions,  M  and  N,  the  width  of  the  ad- 
ditions would  be  determined  by  dividing  496 
by  twice  the  length  of  the  square  already  found, 
2  X  60.     Using  this  as  the  trial  divisor,  we  obtain  4  as  the  width,  which 
is  the  units*  term  of  the  root ;  but  the  entire  length  of  the  additions 
rig.  3.  is  two  times  the  tens,  plus  the  units, 

or   124   (Fig.  3),  the  product  of 
which  by  4,  the  units'  term,  is  496. 
60  +  60  +4         There  being  no  remainder,  4096  is 

found  to  be  a  square  of  which  64  is  the  root,  and  the  length  of  the 
court  is  64  feet. 


s 

60  ft. 

B- 

M 

0 

391.     Examples  in  Square  Root. 


1. 

"What  is  the  square  root  of  841  ? 

Ans.  29. 

2. 

What  is  the  square  root  of  763876  ? 

Ans.  874. 

13. 

What  is  the  square  root  of  13616100  ? 

Ans.  3690. 

4. 

What  is  the  square  root  of  253009  ? 

Ans.  503. 

5. 

What  is  the  square  root  of  1012036  ? 

Ans.  1006. 

0. 

AVhat  is  the  square  root  of  447.3225  ? 

Ans.  21.15. 

.7. 

What  is  the  square  root  of  .005625  ? 

Ans.  .075. 

8. 

What  is  the  square  root  of  .169  ? 

Ans.  .41109+. 

9. 

What  is  the  square  root  of  ^^\  ? 

Ans.  /^ 

27Q 


EVOLUTION. 


10.  What  is  the  square  root  of  ^f  ? 

Note.— 12— A  Ans.  ^. 

11.  What  is  the  square  root  of  lOy^^  ?  Ans.  S^, 

12.  What  is  the  square  root  off?  Ans.  .86602-|-. 

13.  What  is  the  square  root  of  8|?  Ans.  2.8635-1-. 
14  What  is  the  square  root  of  9^9^  ?  Ans.  3.02334-]-. 

15.  What  is  the  square  root  of  f  of  ||-  ?  Ans.  ^^, 

Optional  Examples. 

Note.  — Extract  the  root  in  the  following  to  five  places. 

16.  ^2fxjHf  =  ? 


17.  V21025=:? 

18.  V9801^  =  ? 

19.  V 502681=? 

20.  V22S=? 

21.  V14002564=:? 

22.  V-4X  25  =? 


23.  V2213.7025r=:? 

24.  V^^239 9025  z=  ? 

25.  V2^  =  ? 

26.  V4028049  =  ? 

27.  V20j  =  ? 

28.  V 9569534976  =  ? 

29.  Vl6X752rr:? 


SO.  VgfHI|^  =  ? 

31.  V^of  f  ofH  =  ?    f 

32.  v-144:=:?      i  W- 

33.  V8  =  ? 


34.  V81.10083136==? 

35.  V^=? 

36.  V746841.64r=:? 

37.  VT^gX(6|)^  =  ? 

38.  V769.987=:? 

39.  V^025  =  ? 

40.  V/^  of  .052  =  ? 

41.  V.^  =  ? 


42.  V(.25-^.06l)X(f)2=? 

43.  V 10011  =  ? 


V  S9S,     Practical   Examples. 

1.  There  is  a  field  of  corn  having  an  equal  number  of  rows 
and  hills  in  a  row,  which  contains  1020100  hills  in  all ;  what  is 
the  number  of  rows  in  the  field?  Ans.  1010  rows. 

2.  A  body  of  troops,  consisting  of  2601  men,  has  an  equal  num- 
ber in  rank  and  file  ;  how  many  are  there  in  each  ?  Ans.  51  men. 

3.  A  company  of  persons  spent  $3.24 ;  each  person  spending 
as  many  cents  as  there  were  persons,  how  many  cents  did  each 
spend  ?  Ans.  18  cents. 


SQUARE  ROOT.  271 

4.  "What  is  the  length  of  one  side  of  a  square  farm  containing 
S02  acres,  2  roods  of  land?  Ans.  220  rods. 

5.  What  is  the  length  of  a  square  park  which  contains  2  square 
miles  ?  A)is.  1-4142-|-  miles. 

6.  There  is  a  circular  lot  which  contains  3  acres ;  what  is  the 
length  of  a  square  lot  whose  ai'ea  is  the  same  ? 

Ans.  21.9089+ rods. 

7.  What  is  the  size  of  a  square  lot  whose  area  is  thirty  times 
that  of  the  above?  Ans.  120  rods. 

8.  What  is  the  cost  of  fencing  a  square  lot  which  contains  1 
acre,  at  $5  per  rod?  Ans.  $2 52.9S. 

9.  The  side  of  a  square  is  8  ft.  6  in. ;  what  is  the  side  of  a 
square  having  25  times  the  area  ?  -      ' 

10.  A  owned  a  lot  of  land  51  rods  by  80  rods,  and  another  180 
rods  by  100  rods,  which  he  bartered  with  B  for  a  square  lot  con- 
taining 138  acres;  how  many  rods  less  of  fencing  are  there  in 
the  square  lot  than  in  the  other  two?  Ans.  228  rods  nearly. 

11.  I  have  two  square  lots  of  land,  the  larger  of  which  con- 
tains 270  acres  ;  the  ratio  of  the  smaller  to  the  larger  is  as  5  to 
6  ;  what  is  the  length  of  one  side  of  the  smaller  ? 

Ans.  189.73+ rods^ 

12.  On  a  roof  there  are  laid  5000  slates, — the  number  in  the 
length  being  twice  the  number  in  the  breadth;  wliat  is  the  number 
each  way  ? 

;    Note.  —  It  is  evident  that  the  slates  are  laid  in  two  equal  squares; 
hence  the  square  root  of  ^  of  5000  (V^  of  5000)  will  equal  the  breadth. 
Ans.  50  sla*es  in  breadth  ;  100  slates  in  length. 

13.  Suppose  the  above  roof  to  have  had  10000  slates,  and  the 
breadth  to  have  been  one  third  of  the  length,  what  would  have 
been  the  number  of  slates  in  the  length  and  breadth  ? 

Ans.  173.205+ length;  57.735+ breadth. 

14.  What  is  the  difference  between  the  fencing  of  a  34-acre 
lot,  whether  it  be  a  square  or  a  rectangular  lot,  twice  as  long  as 
it  is  wide?  Ans.  17.89  rods. 

15.  My  orchard  contains  5400  trees ;  the  nupiber  of  trees  in 


272  EVOLUTION. 

width  is  to  the  number  in  lengfli,  as  2  to  3  ;  what  is  the  number 
each  way  ? 

Note.  —  -|  of  the  trees  will  be  a  square,  whose  square  root  will  be  the 
number  of  trees  in  the  width  of  the  orchard. 

16.  Suppose,  in  the  above  orchard,  the  outer  rows  of  trees  to 
stand  upon  the  boundary  line,  and  all  to  stand  30  feet  apart, 
what  is  the  area  covered  by  the  orchard?         Ans.  lOS^^f  acres. 

17^  There  is  a  rectangular  court  paved  with  1728  paving- 
stones  15  inches  square ;  the  length  of  the  court  is  to  the 
width  as  4  to  3  ;  what  is  the  number  of  stones  each  way  ? 

18.  How  many  square  feet  in  the  superficial  contents  of  the 
above  court  ?      ^  ^ 

19.  What  is  the  side  of  a  square  that  will  contain  as  many 
square  feet  as  a  rectangle  w^hose  sides  are  150  and  70  feet  ? 

20.  What  is  the  mean  proportional  between  6  and  24?  (Art.  373.) 

/ 
APPLICATIOK    OF    SQUARE     ROOT    TO    RIGHT-ANGLED    TRI- 
ANGLES 

Definitions. 

39S«  An  Angle  is  the  opening  between  two  lines  that  meet 
each  other. 

394:*  A  Right  Angle  is  the  angle  formed  by  two  lines  that 
are  perpendicular  to  each  other,     (Art.  191.) 

39«>«  A  Triangle  is  a  figure  having  three  angles,  and 
bounded  by  three  straight  lines. 

396.  A  Right-angled  Triangle  is  a  triangle  having  one 
of  its  angles  a  right  angle. 

397.  The  Hypothenuse  of  a  Right-angled  Triangle  is 
the  side  opposite  the  right  angle. 

398.  The  Base  of  a  Right-angled  Triangle  is  the  side 
upon  which  it  is  supposed  to  stand. 

399.  The  Perpendicular  of  a  Right-angled  Triangle 
is  the  side  perpendicular  to  the  base.- 


SQUARE  ROOT 


273 


~ 

VXA 

xy 

X; 

5 

•- 

400.      To    FIND    EITHER    SiDE    OF    A    RiGHT-ANGLED    TRIAN- 
GLE,   THE    OTHER    TWO    SiDES    BEING    KNOWN. 

Suppose  the  figure  A  B  C  to  be  a  right-angled  triangle,  whose 
sides  are  3,  4  and  5  feet  respectively.  A 
square  formed  upon  the  hypothenuse,  A  C, 
will  contain  25  square  feet;  one  formed 
upon  the  base,  B  C,  will  contain  16  square 
feet,  and  one  formed  upon  the  perpendicu- 
lar, A  B,  will  contain  9  square  feet. 
Thus,  it  appears  that  the  square  upon  the 
line  A  C  is  equal  to  the  two  squares  upon 
A  B  and  B  C ;  and  generally, 

The  square  upon  the  hypothenuse  of  a  right-angled  triangle  is 
equal  to  the  sum  of  the  squares  of  the  other  two  sides.     Hence, 

Rule  I.  To  find  the  hypothenuse,  the  base  and  perpendicular 
being  giv^n :  Square  the  base  and  perpendicular,  and  extract  the 
square  root  of  their  sum. 

Rule  II.  To  find  the  base  or  perpendicular,  the  hypothenuse 
and  other  side  being  given  :  Square  the  hypothenuse  and  the  given 
side,  and  extract  the  square  root  of  their  difference. 

4:01.     Examples. 

1.  The  base  of  a  right-angled  triangle  being  30  feet,  the  per- 
pendicular 40  feet,  what  is  the  hypothenuse  ?  Ans.  50  ieet. 

2.  The  hypothenuse  of  a  right-angled  triangle  being  32.5  feet, 
the  base  30  feet,  what  is  the  perpendicular?  Ans.  12.5  feet. 

3.  What  must  be  the  height  of  the  eaves  of  a  house  that  can 
be  reached  by  a  ladder  30  feet  long,  the  foot  of  the  ladder  stand- 
ing 18  feet  from  the  underpinning  of  the  house?      Ans.  24  feet. 

'4.  How  far  from  the  foot  of  a  post  15  feet  high  can  a  horse 
feed  that  has  a  rope  fastened  around  his  neck  and  attached  to  the 
top  of  the  post,  the  distance  being  37  feet  to  the  neck,  and  the 
horse  feeding  two  feet  beyond  the  end  of  the  rope  in  a  direct 
line  with  the  rope  ?  Ans.  36  feet. 

5.  G.  W.  Bailey  had  a  tree,  which  being  partially  broken  off 
18 


274:  EVOLUTION. 

24  feet  from  the  ground,  the  top  struck  the  ground  10  feet  from 
the  foot  of  the  tree,  and  on  a  level  with  it ;  what  was  the  height 
of  the  tree  ?  Ans.  50  feet. 

•  6.  What  must  be  the  length  of  a  ladder  to  reach  to  the  top  of 
a  chimney  48  feet  high,  the  foot  of  the  ladder  being  20  feet  from 
the  chimney  ?  Ans.  52  feet. 

7.  If  the  top  of  the  ladder  mentioned  above  be  lowered  6  feet, 
how  far  will  the  foot  stand  from  the  chimney  ? 

8.  Two  vessels  start  at  the  same  point,  and  sail,  one  due  south 
6  degrees,  and  the  other  due  east  8  degrees ;  how  many  miles 
apart  are  they,  reckoning  69-^  miles  to  a  degree  ? 

9.  What  is  the  width  of  a  street  from  a  point  in  which  a  ladder 
32^  feet  long  will  reach  a  window  26  feet  high  on  one  side,  and 
one  24^  feet  high  on  the  other  side  ?  Ans.  40.85-|-. 

10.  What  is  the  width  of  a  common,  on  which  stands  a  flag- 
staff 195  feet  high,  from  the  top  of  which  to  one  side  of  the 
common  is  675  feet,  and  to  the  other  360  feet  ? 

11.  How  far  from  the  foot  of  a  flagstaff  24  feet  high,  must  a 
ladder  23  feet  long  be  placed  that  a  person  may  ascend  to  within 
5  feet  of  the  top  ? 

/A  12.  My  house  is  40  feet  wide,  and  the  ridge-pole  is  15  feet 
above  the  middle  of  the  beam  which  connects  the  eaves  ;  what  is 
the  length  of  the  rafters  ? 

13.  Provincetown,  Erie,  and  Elmira  are  in  nearly  the  same 
latitude  ;  suppose  Elmira  to  be  243  miles  directly  north  of  Wash- 
ington, Erie  to  be  305  miles  north-westerly,  and  Provincetown 
380  miles  north-easterly,  how  far  is  Provincetown  from  Erie  ?  )  j'  ^' 

14.  Four  persons,  Messrs.  Ames,  Barnes,  Carnes,  and  Doane, 
are  residing  around  Cincinnati,  as  follows :  Ames,  20  miles 
north ;  Barnes,  60  miles  east ;  Carnes,  27  miles  south ;  and 
Doane  36  miles  west  of  the  city ;  what  is  the  shortest  distance 
one  of  these  persons  must  travel  to  visit  all  the  rest,  and  reach 
his  own  home  ?      <'   / 

15.  What  is  the  length  of  the  diagonal,  that  is,  the  distance 
from  one  corner  to  the  opposite  corner,  of  a  square  lot  which 
contains  16  square  rods  ?  Ans.  5.6568-f-  rods. 


SQUARE  ROOT.  275 

16t  The  diagonal  of  the  floor  of  a  square  room  is  110  feet; 
wliat  is  its  area  ?  Ans.  6050  ft. 

17.  The  diagonal  of  a  square  lot  is  75  rods;  what  is  one  side  ? 

18,  What  is  the  diagonal  of  the  floor  of  a  room  which  is  15 
fecit  square  ?    . 

19^  Suppose  the  above  room  to  be  10  feet  high;  what  is  its 
diagonal,  that  is,  its  distiince  from  the  lower  corner  to  the  oppo- 
site upper  corner  ? 

^-f^oTE.  —  The  diagonal  of  the  floor  (Ans.  to  Ex.  18)  becomes  the  base 
of  the  triangle,  of  M'hich  the  diagonal  of  the  room  is  the  hypothenuse ; 
but  the  square  of  the  diagonal  of  the  floor  is  equal  to  the  sura  of  the  squares 
of  the  length  and  width  of  the  room.  Hence,  to  obtain  the  diagonal  of 
the  room,  Square  its  three  dimensions,  and  extract  the  square  root  of  their 
sum, 

20r  What  is  the  diagonal  of  a  cubical  room,  each  of  whose 
dimensions  is  20  feet  ?  Ans.  34.64-|-  feet. 

21*.  What  is  the  diagonal  of  a  room  36  feet  long,  24  feet  wide, 
and  18  feet  high?     Z'  /.^  1/ 

22?  What  is  the  diagonal  of  a  cubical  block  whose  edge  is  2f 
inches?  ^_^ 

23.*  In  the  centre  of  a  square  of  land  containing  one  acre  is  a 
mound  35  feet  high ;  at  the  top  of  this  mound,  which  corresponds 
with  the  centre  of  the  square,  is  a  liberty-pole,  120  feet  high; 
what  is  the  distance  from  the  top  of  the  pole  to  the  nearest  point 
in  the  boundary  of  the  lot? 

24?  What  is  the  distance  to  the  farthest  point  in  the  boundary  J(  /  y^ 

25?  I  have  a  lot  of  land  15  feet  square,  which  I  design  to  ar- 
range in  five  flower-beds  as  follows:  a  central  square  bed,  to 
be  bounded  by  lines  connecting  the  middle  points  of  the  sides  of 
the  original  square,  and  four  equal  triangular  corner  beds,  whose 
sides  extend  5^  feet  from  the  right  angle  at  the  corner;  how 
many  feet  of  bordering  will  be  required  to  surround  all  the 
beds?  ^/25.  117.54  fl.,  nearly. 

26?  At  the  summit  of  a  hill,  which  is  200  feet  in  height,  stands 
a  tower,  20  feet  high  ;  from  the  top  of  the  tower  to  the  foot  of 
the  hill  is  300  feet ;  from  the  top  of  the  tower  to  the  opposita 


276  EVOLUTION. 

side  of  a  stream  which  flows  at  the  foot  of  the  hill  is  400  feet ; 
what  is  the  width  of  the  stream  ? 

S^^  For  Dictation  Exercises,  see  Key. 


)Ie,  showing 

CUBE  ROOT, 

the  third  po 

Eoots, 

1  cubed 

— 

9      « 

zrr 

10      « 

nn 

99      " 

— 

100      " 

— 

999      " 

— 

and  hundreds. 

Cubes. 

i 

729 

iooo 

970299 

1000000 

997002999 

By  the  above  it  will  be  seen  that  the  cube  root  of  any  whole 
number,  composed  of  one,  two,  or  three  figures,  must  be  units ; 
of  four,  five,  or  six  figures,  must  be  units  and  tens ;  of  seven, 
eight,  or  nine  figures,  must  be  units,  tens  and  hundreds ;  and 
hence,  generally,  that  if  we  'point  a  number  off  into  periods  of 
three  figures  each,  heginning  with  the  units,  the  number  of  figures 
in  the  cube  root  will  be  indicated  by  the  number  of  the  periods. 

Note  I.  —  The  left  hand  period  may  contain  but  one  or  two  figures. 

Note  II.  —  The  principle  above  elucidated  applies  to  decimal  frac- 
tions ;  but  every  period  in  decimal  fractions  must  contain  three  figures. 

4:03.  Before  extracting  the  cube  root,  let  us  involve  64  to 
the  third  power,  and  preserve  the  separate  products. 

We  have  already  seen  (Art.  388)  that  the  square  or  second 
power  of  64  is 

602  +  2  X  ((30  X  4)  +  42. 

By  multiplying  this  square  by  64  (60  r|-  4),  we  shall  obtain 
the  third  power  of  64. 


CUBE  ROOT.  277 

642  —  602  +  2  X  ^QQ  X  4)  +  42 

64  =  60  +  4 

60^^2  X  (60  X  60  X  4)  +  (60  X  4^) 
(60^  X  4)  +  2  X  (60  X  4^)  +  4^ 

643  —  60^  +  3  X  (CO2  X  4)  +  3  X  (60  X  4^)  +  4^ 

In  this  product  we  have  four  distinct  parts,  as  follows :  — 
(1),  603,  the  tens  raised  to  the  3d  power,  z=  216000 

(2),  3  X  (602  X  4),  3  X  the  square  of  the  tens  X  the  units,  =:  43200 
(3),  3  X  (60  X  42),  3  X  the  tens  X  the  square  of  the  units,  =  2880 
(4),  43,  the  units  raised  to  the  3d  power,  =:  64 

262144 

Thus  we  see  that  a  cube  whose  root  is  composed  of  tens  and 
units,  contains  (1)  the  cube  of  the  tens,  (2)  three  times  the  square 
of  the  tens  multiplied  by  the  U7iits,  (3)  three  times  the  tens  multi' 
plied  by  the  square  of  the  units,  and  (4)  the  cube  of  the  units. 

4r04:,  Observing,  now,  that  the  first  of  these  parts  is  the 
cube  of  the  tens,  and  that  the  units  is  a  factor  in  each  of  the 
other  parts,  we  will  proceed  to  extract  the  cube  root  in  the 
following 

III.  Ex.     What  is  the  cube  root  of  264609.288 .? 

Operation.  g|  g 

.      .        .     HPH 

264609.288(64.2 

Cube  of  tens, 603=  216 

3  X  sq.  of  tens  (trial  divisor),  .     .      3  X  6O2  =      10800  43509  * 

3  X  tens  X  units, 3  X  60  X  4  =         720 

Square  of  units, 4'  =  16   <■ 

True  divisor, 11536  X  4=  46144 

3  X  sq.  of  tens  (trial  divisor),      .      3  X  6402  =  1228800  2465288 

3  X  tens  X  units, 3  X  640  X  2  =       3840 

Square  of  units, 22  = 4 

True  divisor, 1232644  X  2  =    2465288 

Solution.  Pointing  the  number  oif  into  periods  of  three  figures 
each,  by  placing  a  dot  over  the  units  and  every  third  figure  to  the  right 
and  left,  we  find  that  the  root  will  consist  of  three  figures,  and  the 
cube  of  the  tens  must  be  contained  in  264(000).     The  largest  cube 


278  EVOLUTION. 

contained  in  264(000)  is  216(000),  the  root  of  which  is  6(0) ;  this  we 
write  as  the  tens'  term  of  the  root,  and  subtract  its  cube,  216(000),  from 
264(000),  and  to  the  remainder,  48(000),  bring  down  the  next  period, 
G09. 

We  know  that  this  remainder,  48609,  contains  three  times  the 
square  of  the  tens  (the  term  ah-eady  found),  multiplied  by  the  units  ; 
and  though  it  contains  other  terms,  since  this  is  much  the  largest,  we 
take  3  times  60^  (three  times  the  square  of  the  tens)  for  a  trial  divisor, 
and,  dividing  48609  by  it,  obtain  4  for  the  units'  figure. 

We  multiply  3  X  the  tens  (60)  by  the  new  term  of  the  root  (4), 
and  place  the  product  under  the  trial  divisor,  and  under  this,  place  the 
square  of  the  units'  figure ;  and  thus  form  our  true  divisor,  consisting 
of  the  last  three  parts  of  a  perfect  cube  (Art.  403),  wanting  the  units 
Qs  a  factor  in  each.     Multiplying  their  sum  by  4,  we  have  46144. 

This  we  subtract,  and  to  the  remainder  bring  down  the  next  period. 

Considering  64  as  the  tens  in  the  root,  we  multiply  its  square  (6402) 
by  3  for  a  new  trial  divisor,  and,  proceeding  as  before,  obtain  64.2  foir 
the  cube  root  of  264609.288. 

From  the  above  we  deduce  the  following 

4©S.      KULE  FOR  EXTRACTING  THE  CuBE  EOOT.    Point  off 

the  given  number  into  periods  of  three  figures  eachy  by  placing  a 
dot  over  the  units,  and  every  third  figure  to  the  left  in  whole  nuni' 
bers,  and  to  the  right  in  decimals. 

Find  the  greatest  cube  in  the  left  hand  period,  and  write  its 
root  as  the  first  term  in  the  answer.  Subtract  the  cube  from  the 
left  hand  period,  and  to  the  remainder  bring  down  the  next  period 
for  a  dividend. 

Multiply  the  square  of  the  root  already  found,  considered  as 
tens,  by  three  for  a  trial  divisor.  Divide  the  dividend  by  the  trial 
divisor,  and  place  the  result  as  the  next  term  in  the  root. 

To  the  trial  divisor  add  three  times  the  former  term  in  the  root 
(considered  as  tens),  multiplied  by  the  last  term,  also  the  square 
of  the  last.  Midtiply  this  sum  by  the  last  term,  and  subtract  the 
product  from  the  dividend. 

To  the  remainder  bring  down  the  next  period  for  a  new  div- 
idend. 

Multiply  the  square  of  the  terms  of  the  root  already  found  (consid' 


CUBE  ROOT. 


279 


ere d  as  fens),  hy  three  for  a  trial  divisor,  with  which  divide  and 
proceed  as  before. 

Note  I.  —  When  a  zero  occurs  in  the  root,  annex  two  ciphers  to  the 
trial  divisor,  and,  bringing  down  another  period,  proceed  as  before. 

Note  II.  —  If  a  root  figure  proves  too  large,  substitute  a  lower,  and 
repeat  the  work. 

Note  III.  — AVhen  a  remainder  occurs  after  all  the  periods  are  brought 
down,  the  root  may  be  more  nearly  approximated  by  annexing  periods  of 
zeros,  and  continuing  the  operation. 

Note  1Y.  —  The  cube  root  of  a  common  fraction  may  be  obtained  by 
extracting  the  root  of  each  of  the  terms  when  they  are  perfect  cubes; 
when  they  are  not,  the  fraction  may  be  reduced  to  a  decimal. 

Note  V.  —  Mixed  numbers  may  be  reduced  to  decimal  fractions,  or  to 
improper  fractions  when  the  denominator  of  the  fractional  part  is  a  cube 
number. 


4:06,  The  above  rule  may  be  illustrated  by  means  oir* 
blocks. 

A  cube  number  represents  a  cube,  the  edge  of  which  is  the  cube  root 
of  the  number. 

Let  there  be  a  cube  of  262144  solid  inches,  whose  edge  we  wish  to 
determine. 

Having  found  by  pointing  and  trial  that  the  greatest  cube  of  tens  in 
262144  is  216(000),  the  root  of  which  is  6  tens,  we  will  let  216000 
inches  be  represented  by  the  following  diagram  (Fig.  1),  having  for  its 
edge  6  tens  of  inches,  or  60  inches. 

Subtracting  the  cube, 
216(000)  in.,  from  262144 
inches,  there  will  remain 
46144  inches,  which  may 
be  disposed  on  three  sides 
of  the  cube  already  found, 
so  as  to  retain  the  cubical 
form.  The  square  con- 
tents of  the  addition  upon 
one  side  of  this  cube  will  be  60^  zzr  3600  inches,  and  upon  three  sides 
10800  inches.  Using  this  as  a  trial  divisor,  we  find  the  thickness  of 
the  ad  litions  to  be  4  inches.  The  additions  are  repiesented  by  Fig.  2. 
These  additions  being  made,  the  solid  will  be  represented  by  Fig.  3. 


Fig.  1. 

Fig.  2. 

^^--^y  -™5 

pifiiiiiiiiiiiiiiiiiii 

B 

S 

III 

III 

"iinii 

60  in. 

4i 

n.     4 

in.     4  in 

280 


EVOLUTION. 


Fig.  3. 


Fig.  4. 


Fig.  5. 


Fig.  6. 


To  complete  the  cube,  it  also  requires  three  oblong  rectangular 
blocks,  whose  length  is  60  inches,  and  whose  end  is  4  inches  square 
(Fig.  4)  ;  also  a  cube,  whose  edge  is  4  inches  (Fig.  6).  The  side  of  one 
of  the  oblong  blocks  being  60  X  4,  one  side  of  the  three  will  be  3 
times  60  X  4  zzz  720  square  inches,  and  one  side  of  the  small  cube  will 
be  42  z=  16  square  inches. 

If,  now,  we  multiply  the  sum  of  these  surfaces,  10800  +  720 -f- 16, 
— j:  11536  (Fig.  7),  by  their  thickness,  4,  and  increase  the  cube  216000 

Fig.  7. 


60 

60 

60 

4 

4 

4 

8 

. 

4 

~]4 

3  X  602  -{-  3  X  60  X  4  +  4'-^  =  11536. 


Fig.  8. 


by  the  product,  we  form  a  perfect  cube  (Fig. 
8),  whose  edge  is  64  inches.     And  since  thei-e 
is  no  remainder,  262144  is  a  perfect  cube,  of 
which  64  is  the  root. 


1- 


C41u. 

4:®7,     Examples. 

1.  What  is  the  cube  root  of  2744? 

2.  What  is  the  cube  root  of  24389  ?     . 

3.  What  is  the  cube  root  of  704969? 

4.  What  is  the  cube  root  of  12977875  ? 

5.  What  is  the  cube  root  of  224755712  ? 


Ans.  14. 

Ans.  29. 

Ans.  89. 
Ans,  235. 
Ans.  608. 


CUBE 

ROOT.                                            281 

6.   What  is  the  cube  root  of  122097755681  ?          Ans.  4961. 

7.    What  is  the  cube  root  of  729486108008?          Ans.  9002. 

8.    What  is  the  cube  root  of  19683000  ?                    A7is.  270. 

9.    What  is  the  cube  root  of  195.112  ?                        Ans.  5.8. 

10.    What  is  the  cube  root  of 

.000729  ?                        Ans.  .09. 

11.    W^hat  is  the  cube  root  of  329778750  ?           Ans.  690.8+. 

12.   What  is  the  cube  root  of 

.57? 

Ans.  .8291+, 

13.   What  is  the  cube  root  of 

32^: 

'                       Ans.  3.1854-. 

14.    What  is  the  cube  root  of  4  ? 

Ans.  1.587-4-. 

15.   What  is  the  cube  root  of 

tt¥¥  ?                              ^^s.  j% 

16.    What  is  the  cube  root  of 

A^ 

Ans.  ^. 

17.   What  is  the  cube  root  of 

A\5 

'  (2¥<T  =  t¥^)           ^ns.  f. 

18.    What  is  the  cube  root  of 

4^TT 

?                     Ans.  .1957-1-. 

19.   What  is  the  cube  root  of  1|^  ? 

Am.  1.040O44-. 

^t                             Optional  Exaiviples. 

Note.  —  In  the  following,  the  pupil  need  extract  the  root  to  but  four 

places,  if  decimal  fractions  be  reache 

d. 
30. 

20.     ^1574:64t~? 

■^iii=? 

21.    ^^36926037  =  ? 

31. 

^AWs^? 

22.    ^350402625=? 

32. 

^A¥5=? 

23.     ^y  69224023016=? 

33. 

^if-? 

24.    ^^614255059180216  =  ? 

34. 

^t't  =  ? 

25.     -sy5982325.7=:? 

35. 

^27i  =  ? 

26.     ^525.3425872  =  ? 

36. 

•^y.27=:? 

27.  ^.0009874  =  ? 

28.  ^9575248.5  =:?<7  ^     , 

37. 

^I|  +  25-- 

29.     ^Tq=z? 

58.  (^^125+  V«i9)  —  ^^125  +  819=:.' 

59.  ^283^^=?  J  6/)^ 


40.  5.43  X  194  +  -s^ 27054036008=? 

41.  Find  the  difference  between  the  sum  of  the  cube  roots  of 
13824  and  .000729,  and  the  cube  root  of  their  sum.    ^.   ,,  ;  '/ 


282  EVOLUTION 

4:08,     Practical  Examples. 

1.  "What  is  the  length  of  one  side  of  a  cubical  block  of  granite 
which  contains  7077888  solid  inches  ? 

2.  What  will  be  the  edge  of  a  cubical  pile  of  wood,  composed 
of  1000  loads,  each  8  feet  long,  4  wide,  and  2^  feet  high? 

3.  What  will  be  the  length  of  a  cubical  pile  of  wood  that  will 
contain  one  cord  ? 

4.  What  will  be  the  length  of  a  cube  which  will  contain  ^  as 
much  as  another  whose  edge  is  15  feet?  Arts.  7.5  feet. 

5.  What  is  the  depth  of  a  cubical  cistern  which  will  contain  9 
times  as  much  as  one  whose  depth  is  5  feet? 

Ans.  10.4004-f  feet. 
.  ^.    What  must  be  the  dimensions  of  a  cubical  vessel  that  shall 
contain  300  gallons  of  water,  reckoning  231  cubic  inches  to  a 
gallon?  Ans.  41.075-j-. 

7*  What  will  be  the  cost  of  boards,  at  $11.25  per  thousand 
feet,  to  construct  the  bottom  and  sides  of  a  cubical  bin  which 
shall  contain  75  bushels  of  grain? 

Note.  —  2150.4  cubic  inches  z=:  1  bushel.  Ans.  $1.191-|-. 

8.  What  will  be  the  cost  of  lead,  at  |.12^  per  lb.,  there  being 
1^  lbs.  to  the  s(juare  foot,  to  line  a  cubical  box  containing  15 1 
cubic  feet  ?  C  i  r  ' 

9.  How  many  yards  of  paper,  ^  yard  wide,  will  be  required  to 
line  98  cubical  boxes,  each  containing  5-^^^^  cubic  feet? 

Ans.  384  yards,. 

lOf  The  walls  of  the  ancient  city  of  Babylon  are  said  to  have 

been  350  feet  high,  and  built  of  brick  ;  the  city,  15  miles  square 

inside  the  walls.     Suppose  the  average  thickness  of  the  walls  to 

have  been  GO  feet,  what  M^ould  be  the  length  of  a  cubical  pile 

\,composed  of  the  brick  in  the  walls?  Ans.  1881.2-[-  feet, 

^^  For  Dictation  Exercises,  see  Key. 


MENSURATION. 


283 


MENSURATION. 

409*  The  definitions  of  various  surfaces  and  solids  are 
found  on  pages  109,  112,  114.  Such  as  are  in  general  use,  and 
not  there  found,  are  given  in  this  section. 

PLANE  SURFACES,  RECTILINEAR  FIGURES. 
Triangles. 
410*  The  Right-angled  Triangle  contains  one  right  angle, 
4:11.   The  Obtuse-angled  Triangle   contains  one  obtuse 
angle.   tA^  ^ 


N 

Eight-angled.  Obtuse-angled. 

413.   The  Equilateral  Triangle  contains  three  equal  sides. 

413.  The  Isosceles  Triangle  contains  two  equal  sides. 

414.  The  Scalene  Triangle  has  no  sides  equal. 

M 


N 

Equilateral. 


Scalene. 


Quadrilaterals. 
41^*    A  Parallelogram  is  a  quadrilateral  whose  opposite 
sides  are  parallel. 

416*   A  Rhombus  is  a  parallelogram  whose  sides  are  all 
equal,  and  whose  angles  are  not  right  angles. 

417.   A  Trapezoid  is  a  quadrilateral  only  two  of  whose  sides 
are  parallel. 

M p 


Parallelogram. 


Trapezoid. 


284 


MENSURATION. 


418  •   A  Bectangle  is  a  parallelogram  whose  angles  are 
right  angles. 

4:19.    A  Square  is  a  rectangle  whose  sides  are  all  equal. 

4:^0.    A  Trapezium   is  a  quadrilateral   of  which  no  two 
sides  are  parallel. 


0^ 


Eectantrle. 


Square. 


Trapezium. 


4^1.    The  term  Polygon  is  a  general  name  applied  to  any 
figure  bounded  by  straight  lines. 

42S.  The  Base  of  a  figure  is  the  line 
upon  which  it  is  supposed  to  stand. 

433.  Tlie  Altitude  of  a  figure  is  its 
height.  The  lines  M  N  in  the  preceding 
figures  indicate  altitudes. 

4S4.    The  Diagonal  of  a  figure  is  a 
line  joining  any  two  angles  not  adjacent, 
diagonals. 


Polygon. 

The  lines  O  P  are 


AREAS. 

4^*>,    The  area  of  a  square  or  rectangle  equals  the 
of  its  length  and  its  breadth  or  height.     (Art.  173.) 

4@6.   The  area  of  any  parallelogram  equals  { — 
the  product  of  its  base  and  its  height;  for  it  can 
be  proved  to  be  equal  to  a  rectangle  of  the 
same  base  and  height. 

427.     The  area  of  a  triangle  equals  half^ 
the  product  of  its  base  and  height ;  for  every 
triangle  equals  one  half  of  a   parallelogram 
of  the  same  base  and  height. 


product 


/ 


AREAS.  285 

When  the  three  sides  of  a  triangle  are  given,  the  area  may 
be  found  hy  subtracting  each  side  separately  from  half  the  sum  of 
the  three  sides,  then  multiplying  the  continued  product  of  these 
remainders  hy  half  the  sum  of  the  sides,  and  extracting  the  square 
root, 

4:^8,    The  area  of  a  trapezoid  equals  half  of  the  sum  of  its 

parallel  sides  multiplied  hy  the  distance  between  them  ;  M 

for  it  is  equal  to  two  triangles  whose  bases  are  the 
two  parallel  sides  of  the  trapezoid,  and  whose  altitude 
is  the  distance  between  them.  N 

429.  The  area  of  any  polygon  may  be  found  by  dividing 
it  into  triangles  and  obtaining  the  sum  of  their  areas. 

Note.  — The  student  should  draw  figures  for  each  of  the  following 
problems. 

430.     Examples  for  Practice. 

1.  What  is  the  area  of  a  rectangle  whose  length  is  20  feet  and 
breadth  6j  feet  ?  Ans.  130  sq.  ft. 

2.  How  many  square  feet  of  canvas  in  a  picture  6  ft.  9  in. 
long  and  4  ft.  2  in.  broad  ?  J  &"  *^ 

3.  How  many  square  yards  in  a  garden  20  yards  square  ? 

4.  Required  the  area  of  a  parallelogram  whose  base  measures 
3  ft.  4  in.  and  altitude  1  ft.  3  in. 

5.  Required  the  area  of   a  parallelogram  r -7 

whose  base  measures  23  feet  and  the  adjacent        /_\ / 

side  13  feet,  from  the  extremity  of  which  a         ^ 
perpendicular  drawn  to  the  base  cuts  from  the  base  5  feet. 

Ans,  276  sq.  ft. 

6.  What  is  the  area  of  a  triangle,  the  length  of  the  base  being 
20  feet,  and  the  height  10  ft.  4  in.  ?  Ans,  lO^  sq.  ft. 

7.  What  is  the  area  of  a  right-angled  triangle  whose  base  and 
perpendicular  are  20  and  18  feet  respectively?        /  ^^ 0 

8.  What  is  the  area  of  a  right-angled  triangle  whose  perpen- 
dicular and  hypothenuse  are  42  and  45 J-  feet  respectively  ? 

\f  Ans,  367^  sq.  feet 


266 


MENSURATION. 


9.  Eequired  the  height  and  area  of  an  equilateral  triangle 
whose  sides  are  10  feet  long. 

Ans.  height  8.66+  ft.;  area  43.3+  sq.  ft. 

10.  Eequired  the  area  of  a  triangle  whose  sides  are  3,  8,  and 
10  feet  long  respectively.  A7is.  9.921+  sq.  ft. 

11.  How  many  square  rods  in  a  triangular  lot  of  land  whose 
jsides  measure  14  rods,  32  rods,  and  20  rods,  respectively?        /  i  '(^ 

12.  What  is  the  area  of  a  trapezoid  whose  parallel  sides  are 
14  and  32  feet  long,  and  the  distance  between  them  16  feet? 

Ans.  368  sq.  ft. 

13.  What  is  the  area  of  a  trapezoid  whose  parallel  sides  are 
twice  those  of  the  above,  and  the  distance  between  them  5  ft. 
Tin.?       . 

14.  How  maiiy  sq.  ft.  in  the  surface  of  a  board  which  is  18  ft. 
long,  18  in.  wide  at  one  end,  and  14  in.  wide  at  the  other? 

Ans.  24  sq.  ft. 

15.  How  many  acres  in  a  quadrangular  field  having  2  parallel 
sides  measuring  10  ch.  5  1.  and  16  ch.  8  1.  respectively,  and  the 
distance  between  them  being  15  ch.  ?  Ans.  19.5975  acres. 

16.  What  is  the  area  of  a  trapezium,  the  length  of  a  diagonal 
being  50  feet,  and  of  the  perpendiculars  from  the  opposite  ver- 
tices to  the  diagonal  10  feet  and 
35  feet?  Ans.  1125  sq.ft. 

17.  Find  the  area  of  the  accom- 
panying polygon,  the  dimensions 
being  as  follows  :  A  C,  5  ft. ;  A 
D,  8  ft. ;  A  E,  10  ft. ;  B  M,  3  ft. ; 
C  N,  4  ft.;  DQ,  5  ft.  6  in.;  P  F, 
4  ft.  6  in. 


J^i.h 


CIRCLES. 
4:31*  The  area  of  a  circle  equals  o??e  half  of  the  product 
of  the  circumference  and  radius,  or  one  fourth  of  the  product  of 
the  circumference  and  diameter ;  for  it  may  be  considered  as 
made  up  of  triangles,  whose  bases  compose  the  circumference  ol 
tke  circle,  and  whose  vertices  (Art.  190),  are  at  the  centre. 


CIRCLES.  287 

43^*     Geometricians  have  proved  that  the  circumference  of 
every  circle  is  nearly  3.1416  times  its  diameter.     Hence, 
When  the  Diameter  is  given, 
433*     The  Circumference  =:  Diameter  X  3.1416. 

434.  The  Area  —  (Diametef  X  3.1416)  X  ^^^^  — 
Diameter^  X  .7854. 

When  the  Circumference  is  given, 

435.  The  Diameter  =  «£^^™^ 
When  the  Area  is  given, 

436.  The  Diameter  =:^  1 5I^^ 

\  .7851 

437.     Examples. 

I.  Required  the  circumference  of  a  circle  whose  diameter  is  8 
feet.  Ans,  25.1328  ft. 
^    2.  If  a  radius  is  12  feet,  what  is  the  circumference? 

Ans.  75.398+  ft. 

3.  If  the  circumference  is  100  feet,  what  is  the  length  of  the 
diameter?  Ans.  31.8309+  ft, 

4.  What  is  the  area  of  a  circle  whose  diameter  is  21  feet  ? 

Ans.  346.3614  sq.  ft. 

5.  What  is  the  area  of  a  circle  whose  diameter  is  5  ft.  6  in.  ? 

6.  What  is  the  area  of  a  circle  whose  radius  is  2  ft.  1  in.  ? 

Ans.  13  sq.  ft.  91^  sq.  in. 

7.  What  is  the  area  of  a  circle  whose  radius  is  5  ft.  2  in.  ? 

8.  What  is  the  diameter  of  a  circle  whose  area  is  4  sq.  rods  ? 

Ans.  2.256+  rds. 

9.  What  is  the  radius  of  a  circle  whose  area  is  19  sq.  miles? 

Ans,  2.459+  miles. 

10.  What  is  the  space  occupied  by  a  cart-wheel  -whose  spokes 
are  2  feet  long,  and  the  diameter  of  whose  hub  is  10  inches  ? 

Ans.  18.3478+  sq.  ft. 

II.  How  many  square  yards  in  a  circular  piece  of  cloth  that 
will  cover  a  haycock  measuring  from  the  ground  over  the  top  to 
the  opposite  side  10  feet  ?  Ans.  8.72|  sq.  yds. 


288 


MENSURATION. 


12.  How  many  sq.  inches  in  the  bottom  of  a  square  box  that 
will  contain  a  ring  20  inches  in  diameter  ?  Ans.  400  sq.  in. 

13.  How  many  sq.  inches  in  the  bottom  of  a  square  box  that 
will  be  contained  in  a  circular  box  20  inches  in  circumference  ? 

P  Ans.  20.263  sq.  in. 

14.  How  many  rods  squire  is  a  plat  of  ground  which  contains 
as  much  as  a  circular  plat  that  is  20  rods  across  ? 

Ans.  17.724-f  rds. 

15.  How  many  planks  2  inches  thick  can  be  sawed  from  a  log 
10  feet  in  circumference,  allowing  2  slabs,  each  at  least  3  inches 
thick,  to  be  cast  aside?       •  Ans.  16  planks. 


Solids. 


■P 
Cube. 


Parallelopiped. 


Pyramid. 


Cone. 


Cylinder. 


Frustum  of  a 
Cone. 


Sphere. 


DEFINITIONS,  289 

<i:38»     A  Ctibe  is  a  solid  bounded  by  six  equal  squares, 

439.  A  Parallelopiped  or  Parallelopipedon  is  a  solid 
bounded  by  parallelograms, 

440.  A  Prism  is  a  solid  whose  upper  and  lower  bases  are 
equal  and  parallel  polygons,  and  whose  convex.^urface  is  com- 
posed of  pai-allelograms. 

44 1.  A  Cylinder  is  a  round  body  whose  bases  are  equal 
and  parallel  circles. 

44^.  A  Pyramid  is  a  solid  whose  base  is  a  polygon,  and 
Avhose  convex  surface  is  composed  of  triangles  which  terminate 
in  a  common  point  called  the  Tertex. 

443*  A  Cone  is  a  solid  whose  base  is  a  circle,  and  whose 
convex  surface  tapers  uniformly  to  a  point  called  the  vertex, 

444«  The  Prustum  of  a  Pyramid  or  Cone  is  that  which 
remains  after  cutting  off  the  upper  part  by  a  plane  parallel  to 
the  base, 

44*>»  The  Height  of  any  of  the  solids  here  defined  is  the 
perpendicular  -distance  from  the  highest  point  to  the  base,  (See 
lines  A  B  in  the  preceding  figures.) 

440.    The  Slant  Height  of  a  Kegular  Pyramid  or  of 

a  cone  is  the  shortest  distance  from  the  vertex  to  the  perimeter 
(boundary)  of  the  base.   {See  lines  A  m  in  the  preceding  figures.) 

447.  The  Slant  Height  of  a  Frustum  of  a  Pyramid 
or  Cone  is  the  shortest  -distance  between  the  perimeters  of  the 
two  bases.     (See  lines  o-p  in  the  figures.) 

448,  A  Olobe  or  Sphere  is  a  «olid  l^ounded  by  a  curved 
surface,  every  part  of  which  is  equally  distant  from  a  point 
within  called  the  centre, 

SOLIDITIES  AND   CONYEX  SURFACES. 

449«  The  Solidity  of  a  Parallelopiped  equals  the  product 
of  its  three  dimensions^,     (Art.  178.) 

4^0»  The  Solidity  of  a  Cube  equals  the  cube  of  one  of 
its  edges. 

19 


290  MENSURATION. 

4:51m  T'le  Solidity  of  a  Prism  or  of  a  Cylinder  equals  the 
area  of  its  base  multiplied  by  its  height ;  for  it  is  evident  that  a 
prism  or  cylinder  1  inch  high  must  contain  as  many  cubic  inches 
as  there  are  square  inches  in  the  base ;  and  if  it  is  2,  3,  or  any 
number  of  inches  high,  it  must  contain  2,  3,  or  that  number  of 
times  as  many  solid  inches. 

4:53,  The  Convex  Surface  of  an  Upright  Prism  or  Cyl- 
inder equals  the  perimeter  of  one  of  its  bases  multiplied  by  its 
height;  for  it  is  evident  that,  if  the  prism  or  cylinder  is  1  inch 
high,  its  convex  surface  contains  as  many  sq.  inches  as  there  are 
inches  in  the  perimeter ;  and  if  the  prism  or  cylinder  is  any  num- 
ber of  times  1  inch  in  height,  its  convex  surface  must  contain 
that  number  of  times  as  many  square  inches. 

4:«>3.  The  Solidity  of  a  Pyramid  or  Cone  equals  the  area 
of  its  base  multiplied  by  -^  of  its  height ;  for  it  can  be  proved  that 
these  solids  are  each  ^  of  a  prism  or  cylinder  of  the  same  base 
and  height. 

4^4.  The  Convex  Surface  of  a  Pyramid  or  Cone  equals 
the  perimeter  of  its  base  multiplied  by  ^  of  the  slant  height ;  for 
the  convex  surface  of  each  may  be  regarded  as  composed  of 
triangles  whose  bases  form  the  perimeter  of  the  base  of  the 
solid,  and  whose  height  is  the  slant  height  of  the  solid. 

4.^^*   The  Solidity  of  the  Frustum  of  a  Pyramid  or  Cono 

equals  that  of  three  pyramids  or  cones  whose  bases  are  the  upper 
and  lower  bases  of  the  frustum  and  a  mean  proportional  (Art.  373) 
between  the  two,  and  whose  height  is  the  height  of  the  frustum.. 
Hence,  the  solidity  equals  the  sum  of  the  iivo  bases  plus  the  square 
root  of  their  product,  multiplied  by  ^  of  the  height  of  ttie  fruS' 
tum, 

406.  The  Convex  Surface  of  the  Frustum  of  a  Pyramid 
or  Cone  equals  J-  the  sum  of  the  perimeters  of  the  two  bases  mul- 
tiplied by  the  slant  height ;  for  the  convex  surface  of  each  may 
be  regarded  as  made  up  of  trapezoids  whose  parallel  sides  form  • 
the  perimeters  of  the  bases,  and  whose  height  is  the  slant  height 
of  the  frustum. 


SOLIDITIES  AND  CONVEX  SURFACES.  291 

4l57»  Geometricians  have  proved  that  the  Convex  Surface 
of  a  Sphere  equals  the  circumference  multiplied  hy  the  diameter, 
or  equals  the  area  of  four  great  circles  *  of  the  sphere, 

458.  The  Solidity  of  a  Sphere  is  equal  to  its  surface  mul- 
tiplied hy  ^  of  the  radius,  or  -i  of  the  diameter ,  for  the  sphere  may 
be  regarded  as  made  up  of  pyramids  whose  bases  comprise  the 
surface  of  the  sphere,  and  whose  vertices  are  at  the  centre. 

From  the  preceding  explanations,  and  by  the  use  of  the  well 
estalDlished  fact  that  the  circumference  of  every  circle  is  3.1416 
times  the  diameter,  the  following  formulas  for  finding  the  solid 
contents  and  convex  surfaces  of  cylinders,  cones,  frustums  of 
cones,  and  spheres,  are  obtained. 

To  save  space,  D  will  be  used  for  diameter  of  lower  base,  D' 
for  diameter  of  upper  base,  h.  for  height,  and  s.  h.  for  slant 
height. 

459.  The  SoUd  Contentsof  a  Cylinder  =  D2  x  .7854Xh. 

460.  The  Solid  Contents  of  a  Cone  =  D^  X  .7854  X-- 

461.  The  Solid  Contents  of  a  Frustum  of  a  Cone  = 
(D2  X  .7854  +  D'2  X  .7854  +  D  x  D'  X  .7854)  x\  = 
(D2  -I-  D'2  -f  D  X  D')  X  .7854  x|- 

46^.   The  Convex  Surface  of  a  Cylinder  =  D  X  3.1416 
X  h. 
46S.   The  Convex  Surface  of  a  Cone  =  D  X  3. 1416  X  — 

464.  The  Convex  Surface  of  a  Frustum  of  a  Cone  = 
(D  X  3.1416  +  D'X  3.1416)  X  '^•• 

465.  The  Convex  Surface  of  a  Sphere  =  D  X  3.1416  X 
D  =  D2  X  3.1416. 

.5236 

466.  The  Solid  Contents  of  a  Sphere  =  B^  x  ^.t4.t<^ 

X-j=D3x.5236. 

*  A  great  circle  of  a  sphere  is  a  circle  which  divides  the  sphere  into 
two  equal  parts. 


292  MENSURATION. 

467.     Examples. 

-1.   How  many  cubic  feet  does  a  block  of  granite  contain,  that 

is  12  feet  long,  4  feet  wide,  and  1^  feet  thick  ?      Ans.  72  cu.  feet. 

2.  What  number  of  cubic  feet  are  there  in  a  cube  whose  edge 
is  1  foot,  11  inches  ?  Ans.  7.041-|-  cu.  feet. 

3.  How  many  cubic  feet  in  a  prism  whose  base  is  a  parallelo- 
gram 15  feet  long  and  4  feet  wide,  and  whose  height  is  9  inches  ? 
W  1^  /  W^f  ^'^</  ^O  -7/^^- ^^^n-;       Ans.  45  feet. 

-it.  Required  the  contents  of  a  prism  whose  base  contains  8^ 
square  yards,  and  the  square  of  whose  height  equals  3  times  the 
number  of  square  feet  in  the  base.  Ans.  41  f  cu.  yards. 

.-  b.  Required  the  contents  of  a  pyramid  whose  base  is  the  same 
as  the  above,  and  whose  height  is  5  feet. 

Ans.  4  cu.  yards,  17  cu.  feet. 

6.  Required  the  contents  of  a  pyramid  whose  base  is  7  feet 
square,  and  whose  height  equals  the  diagonal  of  the  base. 

Ans.  161.69+  cu.  feet. 

7.  Required  the  contents  of  the  frustum  of  a  pyramid  whose 
bases  are  12  and  JOS  square  feet,  and  whose  height  is  18  feet. 

/  ^  ^;v  Ans.  936  cu.  feet. 

8.  What  is  the  convex  surface  of  a  prism,  the  perimeter  of 
whose  base  is  7  yards,  2  feet,  and  whose  height  is  5  yards,  1  foot  ? 

Ans.  40 1  sq.  yards. 

9.  Required  the  number  of  square  feet  in  the  surface  of  a 
four-sided  pyramidal  roof,  the  length  of  each^side  b«"ing  20  fpet^ 
and  the  slant  height  18  fect.x^"  <  /  r  =  i^  ^  7  ^m.hi()%^.  feet. 

10.  What  would  be  the  square  contents  of  a  four-sided  pyram- 
idal roof,  the  length  of  each  side  being  48  feet,  and  the  highest 
point  10  feet  above  the  eaves?  Ans.  2496  sq.  feet. 

11.  Required  the  number  of  square  feet  in  the  sides  of  an  oc- 
tangular (eight-sided)  tower,  the  length  of  each  side  of  the  base 
being  2  feet,  9  inches,  that  of  each  side  of  the  top  1  foot,  1 D 
inches,  and  the  height  of  the  tower  to  the  roof,  measured  on  the 
side  12  feet.  Ans.  220  sq.  feet. 

12.  Required  the  capacity  of  a  cylindrical  cistern,  measuring 
6  feet  across  and  8  feet  deep.  Ans.  226.1 95-|-  eu.  feet. 


76.y>^H=ljL)  £/, 


-•,-:l.(l  I- 


RELATIONS  OF  CIRCLES. 


293 


t. 


^3.  Required  the  capacity  of  a  conical  pit,  measuring  8  feet 
across  and  5  feet  from  the  edge  to  the  deepest  part. 

Ans.  50.2656  cu.  feet 

14!  How  many  quarts  of  water  will  a  circular  tin  pan  contain, 
that  measures  across  the  bottom  11^  inches,  across  the  top  14 
inches,  the  slant  height  being  3^  inches  7         Ans.  6.65-(-  quarts. 

15.  How  many  cubic  feet  in  a  ball  5  feet  in  diameter  ? 

Ans.  65.45  cu.  feet. 

16.  How  many  square  feet  in  the  surface  of  the  ball  ? 

Ans.  78.54  sq.  feet. 

17.  How  many  square  inches  of  leather  will  cover  a  ball  4 
inches  in  diameter  ? 

18.  What  proportion  do  the  cubic  contents  of  a  cone  bear  to 
the  contents  of  a  cylinder  which  will  just  contain  it  ?  Ans.  \, 

19.  Wbat  proportion  do  the  cubical  contents  of  a  sphere  bear 
to  the  contents  of  a  cylinder  which  will  just  contain  it?       Ans.  §. 

20*.    Suppose,  when  the  moon  is  238600  miles  from  the  earth,t 

that  its  shadow  just  reaches  the  earth's  surface,  how  many  cubic 

miles  in  the  shadow,  allowing  the  diameter  of  the  moon  to  be 

21(5(X  utiles,  and  that  of  the  eai^th  to  be  8000  miles  ? 

'^^"^^  •K-  4  I  /  ^  ^- ^A4.483,914,786,355.2  cu.  miles. 

'^RELATIONS   OF   61RCLES;  STtMILAR    TRtANGLES,   AND' POLY- 

4:G8.  It  will  be  apparent, 
by  the  annexed  diagrams,  that 
a  figure  1  inch  square  will  con- 
tain 1  square  inch,  one  2  inches 
square  will  contain  4  square 
inches,  one  3  inches  square  will 
contain  9  square  inches,  and 
thus,  generally,  that  the  areas 


J 


GONS. 


3  in.  square. 


2  in.  square 

lin. 

D 

1  sq. in.        4  sq. in. 


9  sq. in. 


of  squares  are  to  each  other  as  the  squares  of  their  edges. 


t  The  distance  is  measured  from  the  centre  of  the  earth  to  the  centre  of 
the  moon. 


294  MENSURATION. 

The  same  principle  applies  to  circles,  triangles,  and  all  figures  that 
are  similar  to  each  other;*  hence, 

4:69,  I.  The  Areas  of  Similar  Triangles  and  Polygons 
are  to  each  other  as  the  squares  of  their  corresponding  dimen- 
sions. 

III.  Ex.  A  triangle  whose  base  is  10  feet  has  an  area  of 
15  feet ;  what  is  the  area  of  a  similar  triangle  whose  base  is  12 
feet? 

By  Proportion,  102  ;  122—  15  ;  2I.6  square  feet,  Ans. 

470,  II.  The  Areas  of  Circles  are  to  each  other  as  the 
squares  of  their  diameters,  semi-diameters,  and  circumferences. 

III.  Ex.  If  a  pipe  of  2  inches  diameter  will  empty  a  cis- 
tern in  3  hours,  what  must  be  the  diameter  of  a  pipe  to  empty 
the  same  cistern  in  1  J-  hours  ? 

By  Proportion,  1 J  :  3  =:  22  :  8,  the  square  of  the  diameter  of  the  re- 
quired pipe.  v^8  zz=  2.828  +  inches,  A/is, 

471*     Examples. 

~l.  If  the  pot  to  a  furnace  which  consumes  60  lbs.  of  coal  a  day 
IS  24  inches  in  diameter,  what  amount  of  coal  will  be  consumed 
in  the  same  time  by  a  furnace  whose  pot  is  15  inches,  all  other 
conditions  being  the  same  ?  Ans.  23.437-|-  lbs. 

2.  If  a  rope  3  inches  in  diameter  weighs  20  lbs.,  what  is  the 
diameter  of  a  rope  of  the  same  length  which  weighs  9  lbs.  ? 

Ans.  2.012-fin. 

3.  If  a  pipe  4  inches  in  diameter  fills  a  cistern  in  20  minutes, 
15  seconds,  in  what  time  will  a  pipe  that  is  2 J-  inches  in  diameter 
fill  the  same  cistern?  Ans.  51.84  minutes. 

H    4.  If  it  costs  $10.50  to  cover  a  roof  whose  length  is  7  feet,  what 
"will  it  cost  to  cover  a  similar  roof  whose  length  is  21  feet? 

Ans.  $94.50. 

*  Angular  figures  are  similar  when  their  angles  are  equal,  and  their 
corresponding  sides  proportional ;  and,  conversely,  similar  figures  hare 
their  corresponding  sides  proportional. 


SIMILAR  SOLIDS.  295 

5  The  hypothenuse  of  a  right-angled  triangle  is  40  feet ;  what 
fliust  be  the  hypothenuse  of  a  similar  triangle  that  it  may  contain 
twice  the  area?  Ans.  5Q.5QS-{-. 

6.  If  a  circular  lot  of  land  which  is  10  rods  in  diameter  con- 
tains 78.5398  square  rods,  what  number  of  rods  will  a  lot  contain 
which  is  o  rods  in  diameter  ?  Ans.  19.63495. 

V7.  The  area  of  a  triangle  whose  base  is  24  feet  is  120  feet; 
what  is  the  area  of  a  similar  triangle  whose  base  is  96  feet  ? 

Ans.  1920  feet. 

•^'    8.  The  Winchester  bushel  is  18|-  inches  in  diameter  and  8 

inches  deep  ;  what  must  be  the  diameter  of  a  circular  measure  6 

inches  deep,  that  it  may  hold  a  bushel  ?  Ans.  21.36-|-  inches. 

9.  I  have  a  circular  flower-garden,  the  circumference  of  which 
is  boi^ered  with  75  yards  in  length  of  sodding ;  how  many  yards 
will  be  required  to  border  a  circular  garden  of  f  the  area  ? 

Ans.  61.237-f-  yards. 

10.  Having  a  triangular  board  7  J  feet  long,  what  distance  from 
the  base  end  shall  I  cut  it  to  divide  it  into  two  equal  parts  ? 

J[ws.  2.197  — ft. 


SIMILAR  SOLIDS. 

"Note.  —  Angular  solids  are  similar  when  their  angles  are  equal  each 
to  each,  and  arranged  in  the  same  way,  and  their  corresponding  edges 
proportionaL 

The  following  proposition  may  be  easily  proved  by  geom- 
etry :  — 

473o  The  Solidities  of  Cubes,  Spheres,  a7id  all  Similar 
Solids  are  to  each  other  as  the  cubes  of  their  corresponding 
dimensions. 

III.  Ex.,  I.  How  man}'-  globes  of  G  inches  diameter  can  be 
made  from  a  globe  of  48  inches  diameter  .^ 

By  Proportion,  6^  :  48^=:  1  (globe)  :  512  globes,  Ans, 

III.  Ex.,  II.  If  a  conical  stack  of  hay  which  contains  f  of  a 
ton  is  6  feet  high,  what  is  the  height  of  a  similar  stack  which 
contains  3f  tons  ? 


296  MENSURATION. 

By  Proportion, 

I  :  3f  =  63  :  1728,  the  ctibe  of  the  height  of  the  larger  stack. 
^1728=  12  ft.,  height  of  larger  stack,  Ans. 

Examples. 

1.  If  an  ounce  ball  is  |  inch  in  diameter,  how  many  ounce  balls 
can  be  made  from  a  globe  of  lead  G  inches  in  diameter  ? 

An&.  SUj%%  balls. 

2.  A  pyramid  which  is  0  feet  in  height  contains  48  cubic  feet ; 
what  is  the  height  of  a  similar  pyramid  that  contains  100  cubic 
feet?  Ans,  UA9i-\- ieeL 

3.  If  a  cube  of  granite  whose  edge  is  2  feet  weighs  1336.32 
pounds,  what  will  be  the  weight  of  a  cube  whose  edge  is  4  feet, 
9  inches?  Ans.  17901.99. 

4.  If  an  egg  of  2|  inches  in  eircumference  weigh  1  ounce,  what 
would  another  of  the  same  form  and  consistency  weigli  whose 
circumference  is  6  inches  ? 

5.  What  must  be  the  height  of  a  cone  t&  contain  I2i1  times  as 
many  solid  inches  as  a  similar  cone  3  inches  in  height  ? 

An»,  15  inches. 

6.  If  a  bushel  measure  is  18^  inches  in  diameter  and  8  inches 
deep>  what  must  be  the  diameter  and  depth  of  a  half-bushel 
measure  similar  in  form  ? 

Ans,  Diam.  14.683+  in.;  depth  6.349+  in. 

7.  If  an  elephant^s  tusk  9^  feet  long  and  8  inches  in  diameter 
at  base  weigh  214  pounds,  what  would  be  the  dimensions  of  a 
similar  tusk  weighing  75  pounds  ?    (    >,,    /     I 

8.  Estimating  the  mean  diameter  of  the  earth  at  7912  miles, 
and  that  of  the  moon  at  2160  miles,  how  many  bodies  of  the  size 
of  the  moon  could  be  made  fr^m  the  bulk  of  the  earth  ? 

9.  If  the  bulk  of  Saturn  be  1000  times  as  great  as  that  of  the 
earth,  what  is  the  diameter  of  Saturn  f  / 

10.  At  what  distance  from  the  top,  parallel  with  the  base,  must 
a  conical  sugar-loaf  12  inches  high  be  cut  that  it  may  he  divided 
into  two  equal  parts  ? 

11.  Mr.  Hoot  has  three  stacks  of  hay  of  similar  shape,  the 


QUESTIONS  FOR  REVIEW.  297 

* 

diameters  of  their  bases  being,  respectively,  10,  12,  and  14  feet; 
if  the  one  whose  diameter  is  10  feet  contains  2^  tons,  what  will 
each  of  the  others  contain  ?    ^/     ^  §  C    ^  !o 

l^  For  Dictation  Exercises,  see  key.  ^ 

4TS.     Questions   for  Review. 

What  is  Involution.  ?  a  power  ?  an  exponent  ?  What  is  the  first 
power  ?  second  power  ?  third  power  ?  fourth  power  ? 

What  are  the  second  and  third  powers  generally  called  ? 

Rule  for  Involution  ?  How  may  a  mixed  number  be  raised  to  a  given 
power  ?  Repeat  the  squares  of  the  integers  from  1  to  25  ;*  the  cubes 
of  the  integers  from  1  to  10.* 

How  does  Evolution  differ  from  Involution  ? 

What  is  a  root  ?  What  is  the  square  root  of  a  number  ?  the  cube 
root  ?  How  is  the  square  root  indicated  ?  the  cube  root  ?  How  other- 
wise may  the  root  of  a  number  be  indicated  ? 

If  a  power  contain  one  or  two  figures  only,  of  how  many  figures  will 
its  square  root  consist  ?  If  a  power  contain  three  or  four  figures  ?  If 
five  or  six  ? 

What  three  terms  does  every  square  number  contain  whose  root 
consists  of  tens  and  units  ? 

Give  the  rule  for  extracting  the  square  root.  Of  how  many  figures  may 
the  left-hand  period  in  whole  numbers  consist  ?  of  how  many  must  every 
period,  except  this,  consist  ?  of  how  many,  every  period  in  decimals  ? 
''How  do  you  proceed  when  a  zero  occurs  in  the  root ?  how  when  a 
root  figure  proves  too  large  ?  how  when  there  is  a  remainder  ?  How 
do  you  extract  the  square  root  of  a  common  fraction  whose  terms  are 
square  numbers  ?  whose  terms  are  not  squares  ?  How  extract  the  root 
of  a  mixed  number  ?  Explain  the  extraction  of  the  square  root  by  an 
example.     Illustrate  by  diagrams. 

/"What  is  an  angle  ?  a  right  angle  ?  a  triangle  ?  a  right-angled  trian- 
gle ?  its  hypothenuse  ?  its  perpendicular  ?  its  base  ? 

To  what  is  the  square  on  the  hypothenuse  of  a  right-angled  triangle 
equal?     Rule  to  find  the  hypothenuse ;  to  find  base  or  perpendicular. 

If  a  cube  number  contain  one,  two,  or  three  figures  only,  of  how 
many  figures  will  its  cube  root  consist  ?  if  it  contain  four,  five,  or  si^ 
figures  only? 

To  what  four  terms  is  every  cube  number  equal  whose  root  consist! 
of  tens  and  units  ? 

*  At  the  option  of  the  teacher. 


298  QUESTIONS  FOR  REVIEW- 

♦ 

Give  t*he  rule  for  extracting  the  cube  root  Of  how  many  figures 
may  the  left-hand  period  consist?  of  how  many  must  every  other 
period  consist  in  whole  numbers  ?  in  decimals  ? 

How  do  you  proceed  when  a  zero  occurs  in  the  root  ?  how  when  a 
root  figure  proves  too  large  ?  how  when  there  is  a  remainder  ?  How 
do  you  extract  the  cube  root  of  a  common  fraction  when  the  terms  are 
cubes'?  how  when  the  terms  are  not  ?  How  extract  the  cube  root  of  a 
mixed  number  ? 

Explain  the  extraction  of  the  cube  root  by  an  example.  Illustrate 
by  blocks. 

What  is  Mensuration  ?  Name  and  describe  the  different  kinds 
of  triangles.  Draw  a  right-angled  triangle  ;  an  obtuse-angled  triangle ; 
an  equilateral  triangle  ;  an  isosceles  triangle ;  a  scalene  triangle.  Name 
and  describe  the  different  kinds  of  quadrilaterals.  Draw  a  square ;  a 
rectangle  ;  a  rhombus  ;  a  trapezoid ;  a  trapezium  ;  a  circle ;  a  polygon 
of  5  sides  with  2  diagonals. 

'   How  do  you  find  the  area  of  a  square  ?  a  rectangle  ?  any  parallelo- 
gram ?  a  triangle  ?  a  trapezoid  ?  a  trapezium  ?  any  polygon  ? 

How  do  you  find  the  circumference  of  a  circle  when  the  diameter 
is  given?  when  the  radius  is  given?  How  do  you  find  the  area 
of  a  circle  when  the  diameter  is  given?  when  the  radius  is  given? 
How  do  you  find  the  diameter  when  the  circumference  is  given  ?  How 
find  diameter  when  the  area  is  given  ?  How  do  you  find  the  radius 
when  the  area  is  given  ? 

Define  a  cube ;  parallelopiped  ;  prism  ;  cyhnder ;  pyramid ;  cone ; 
frustum  of  a  pyramid  or  cone ;  a  sphere.  Draw  or  mention  some- 
thing in  the  form  of  each  of  these  solids.  What  is  the  height  of  any 
solid  ?  the  slant  height  of  a  pyramid  or  cone  ?  the  slant  height  of  a 
frustum  of  a  pyramid  or  cone  ? 

How  do  you  obtain  the  solid  contents  of  a  cube  ?  a  parallelopiped  ? 
a  prism  or  cylinder  ?  a  pyramid  or  cone  ?  a  frustum  of  a  pyramid  or 
cone  ?     How  do  you  find  the  convex  surface  of  each  of  these  solids  ? 

When  the  diameters  and  altitude  are  given,  how  do  you  find  the 
solid  contents  of  a  cylinder  ?  of  a  cone  ?  of  a  frustum  of  a  cone  ?  of  a 
sphere  ?  How  do  you  find  the  convex  surface  of  a  cylinder  ?  of  a  cone  ? 
the  frustum  of  a  cone  ?  a  sphere  ? 

What  proportion  exists  between  the  areas  of  squares  ?  of  circles  ?  of 
all  similar  triangles  and  polygons  ?     When  are  angular  figures  similar  ? 

What  proportion  exists  between  spheres  ?  between  all  similar  solids? 
When  are  angular  solids  similar  ? 


REVIEW.  299 

474.     General  Review,  No.  8. 

Pl.  Supply  the  2d  term  ia  the  proportion  3| :  ?  =  8  :  25  X  G^/  2 
2.  Wliat  is  the  mean  proportional  between  .8  and  .72  ?  ;  J  .^v^^ 
/      3.  Divide  $1900  between  two  men,  in  the  proportion  of  3  to  5.1  ^ 

4.  Divide  $45  between  three  boys,  so  that  one  shall  have  as 
much  as  the  other  two.  whose  shares  are  as  2  to  7.  y^J 

5.  If  15  gallons  of  oil  cost  7  £,  10  s.,  what  will  25  J-  gallons  cost  ?     .5 
"^.  .6.  How  many  pounds  can  5  horses  draw,  if  6  horses  can  draw 

as  much  as  10  oxen,  and  2  oxen  can  draw  2400  pounds?  •. 

7.  Smith  and  Lee  formed  a  partnership.     Smith  put  in  $1000 
for  6  months,  and  Lee  $600  and  his  services  for  8  months,  his 
services  being  equal  to  $100  a  month.    They  gained  $1506  ;  what 
was  each  one's  share  ? 
-—  8.  What  is  the  5th  power  of  23  ?  the  cube  of  96  ? 

9.  What  is  the  largest  number  of  men  in  a  regiment  of  1000 
that  can  be  arranged  in  a  squarer;  and  how  many  men  will  re- 
main?  How  many  men  will  there  be  on  each  side  of  the  square  ? 

10.  How  many  feet  of  fencing  around  a  square  farm  containing 
15  acres? 

11.  A  ladder  27f  feet  long  reaches  a  window  25f  feet  from 
the  ground ;  how  far  does  the  foot  of  the  ladder  stand  from  the 
house?  ,4        :'  /  "■ 

^12.  Required  the  diameter  of   a  circular  grass  plat  which 
contains  314^^  square  feet. :. 

13.  How  many  rods  of  fencing  on  both  sides  of  a  road  which 
surrounds  a  circular  park  containing  |  of  a  square  mile,  the  road 
being  3  rods  wide  ? 

14.  What  must  be  the  depth  of  a  pail,  that  is  10  inches  across, 
to  contain  5  gallons  (the  sides  being  upright)  ? 

15.  How  many  feet  of  canvas  are  required  to  construct  a  con- 
ical tent  14  feet  across  the  bottom  and  9^  feet  from  the  highest 
point  to  the  ground  ? 

16!  How  many  gallons  will  a  circular  vat  contain,  which  meas- 
ures across  the  top  8  ft.,  across  the  bottom  7  ft.,  the  sides  sloping 
uniformly'and  measuring  on  the  slope  6J-  ft.  in  depth?       ; 
^^  For  changes,  see  Key.  /  'V  ^  f  \  /  '^ 


300  ALLIGATION. 


ALLIGATION. 

^^5»  Alligation,  or  Average,  treats  of  the  mixing  of  dif* 
ferent  ingredients. 

4:76.  Alligation  Medial  is  the  process  of  determining  the 
average  or  mean  value  of  given  quantities  of  different  values. 

4:77.  Alligation  Alternate  is  the  process  of  determining 
what  quantities  of  different  values  may  be  so  combined  that  the 
mixture  shall  be  of  a  given  value. 

Note.  — The  word  alligation  means  a  tyinp  together,  and  is  applied  to 
these  processes  because,  in  the  solutions  of  many  examples,  the  amounts 
or  prices  of  articles  are  linked  or  tied  together.  Average  is  perhaps  the 
better  name  to  use,  as  it  applies  to  all  the  examples. 

ALLIGATION  MEDIAL, 

478.  III.  Ex.  Let  it  be  required  to  mix  10  lbs.  of  sugar 
at  7  cents  per  lb.  with  7  lbs.  at  9  cents,  and  8  lbs.  at  11  cents ; 
what  will  be  the  value  of  the  mixture  ? 

Operation.  The  price  of  10  lbs.  at  7  cents  per 

10  X    7  =  70  lb.'=  70  cents;  of  7  lbs.  at  9  cents 

7  X    9  ziz  63  =63  cents  ;  of  8  lbs.  at  11  cents  = 

8  X  11  =  88  88  cents.   Adding,  we  find  the  value 
25            )221  of  the  mixture  to  be  221  cents,  and 

"^  cents,  Ans,      *^^  ^"°^^^*  °^  P«""^^'  ^«  ^'  -^'    ^^ 
^°^  25  lbs.  are  worth  221  cents,  1  lb.  is 

worth  -^  of  221  cents  =:  8  ^2^1^  cents,  Ans.     Hence  we  deduce  th^  fol- 
lowing 

Rule.  To  find  the  mean  value  of  given  quantities  of  different 
values :  Divide  the  sum  of  the  values  of  the  several  quantities  hy 
the  sum  of  the  quantities. 

Examples. 

1.  If  10  lbs.  of  raisins  worth  10  cents  per  lb.  be  mixed  with  4 
lbs.  worth  15  cents  per  lb.,  what  is  the  value  of  the  mixture  per 
pound?  Ans.  11^  cents. 


ALLIGATION  MEDIAL.  301 

^  2.  There  are  in  a  certain  school,  10  pupils  14  years  old;  9  pu- 
pils 12  years  old;  5,  11  years;  8,  9  years,  and  17,  10  years  old; 
what  is  their  average  age  ? 

"^■^3.  A  family  spent,  during  the  year,  as  follows :  in  January 
$89.75,  in  February  $70.16,  in  March  185.32,  in  April  $90.21, 
in  May  $87.00,  in  June  $66.14,  in  July  $69.42,  in  August  $72.68, 
in  September  $80.65,  in  October  $90.45,  in  November  $98.54, 
in  December, $109.63  ;  what  was  their  average  expense  per 
month?         4-.-'5t  V     ■    "^ 

4.  In  Philadelphia,  during  the  year  1861,  rain  or  snow  fell  as 
follows  :  in  January  on  13  days,  in  February  on  9  days,  in  March 
on  9,  in  April  on  9,  in  May  on  13,  in  June  on  15,  in  July  on  14, 
in  August  on  12,  in  September  on  6,  in  October  on  10,  in  No- 
vember on  11,  in  December  on  4  ;  what  was  the  a\|erage  number 
of  days  per  month  when  rain  or  snow  fell  ?     I    v     j-^ 

5.  In  Massachusetts,  during  the  year  1850,  the  value  of  home 
manufactures  was  $205,333.  During  the  year  1860,  it  was 
$245,886.  What  M-as  the  average  rate  of  increase  per  year 
during  the  10  years  ?      q  (J  b^-/:>\J  P 

6.  A  flour  merchant  sold  50  bbls.  flour  at  $7.50  per  bbl.,  60 
bbls.  at  $9.00  per  bbl.,  25  bbls.  at  $8.50,  40  at  $8.75,  and  100,  at 
$9.50;  what  did  his  sales  average  per  barrel.'*       V    ':  \  '•-,'{    )) 

7.  A  baker  made  wedding-cake  of  the  following  ingredients : 
5  lbs.  flour  worth  5  cents  per  lb.,  5  lbs.  sugar  at  11  cents  per  lb., 
5  lbs.  of  butter  at  22  cents  per  lb.,  6  lbs.  raisins  at  17  cents  per 
lb.,  12  lbs.  currants  at  20  cents,  per  lb.,  2  lbs.  citron  at  50  cents 
per  lb.,  50  eggs,  l^lbs.  to  the  dozen,  18  cents  per  dozen,  ^  pint 
wine  at  374  cents  per  pint,  3  oz.  cinnamon  at  56  cents  per  lb., 
3  oz.  nutmegs  at  $1.00  per  lb.,  1^  oz.  mace  at  $1.00  per  lb.  Al- 
lowing $2.00  for  labor  and  fuel,  -^  lb.  for  the  weight  of  the  wine, 
and  1  oz.  in  every  lb.  for  loss  of  weight  in  baking,  what  was  the 
cost  of  the  cak«  per  lb.  ?  Ans.  $.24^^*  |  ^. 


302 


ALLIGATION. 


V. 


ALLIGATION   ALTERNATE. 


479.  III.  Ex.  A  merchant  has  teas  of  the  following  val- 
ues per  lb.,  42,-  68,  75,  and  84  cents,  with  which  he  wishes  to 
make  a  mixture  worth  70  cents  per  pound.  How  many  pounds 
of  each  kind  shall  he  take  ? 


70 


42  -{-  28- 
68 -|-    2-| 


Operation. 

lib. 
5  lbs. 
S-"     2  lbs. 
14-^  2  lbs. 
A71S.  1  lb.  at  42,  5  lbs.  at  68, 
2  lbs.  at  75,  and  2  lbs.  at  84 
cents. 


We  first  compare  the  various 
prices  of  the  tea  with  the  price  ot 
the  mixture.  If  that  which  is  worth 
42  cents  is  sold  at  70  cents,  there  is 
a  gain  of  28  cents  on  one  pound, 
which  we  indicate  by  writing  -(-28 
opposite  42.  In  the  same  way  we 
find  there  is  a  gain  of  2  cents  per 
lb.  on  the  68-cent  tea,  a  loss  of  5 
cents  per  lb.  on  the  75-cent  tea,  and  a  loss  of  14  cents  per  lb.  on  the  84- 
cent  tea.  We  indicate  the  gain  and  losses  by  their  proper  signs,  and 
proceed  to  take,  two  by  two,  such  kinds  of  tea  and  of  such  quantities 
that  the  gains  shall  balance  the  losses.  Comparing  the  first  with  the 
fourth,  we  find  that  the  gain  on  1  lb.  of  the  first  equals  the  loss  on  2  lbs. 
of  the  fourth.  We  also  find' that  the  gain  on  5  lbs.  of  the  second  equals 
the  loss  on  2  lbs.  of  the  third.  We  therefore  take  1  lb.  of  the  first,  5 
of  the  second,  2  of  the  third,  and  2  of  the  fourth  ;  or,  we  may  take 
any  quantities  of  the  first  and  fourth  that  are  in  the  ratio  of  1  to  2,  and 
of  the  second  and  third  that  are  in  the  ratio  of  5  to  2. 

Instead  of  comparing  the  first  with  the  fourth,  and  the  second  with 
the  third,  we  may  compare  the  first  and  third  and  the  second  and  fourth 
together,  thus :  — 


70 


42  +  28--,  5  lbs. 

68  -f-   2-f -1  7  lbs. 

75  _   5-1  I  28  lbs. 

84  —  14 — ^  1  lb. 


Other  comparisons  might  be  made,  and  thus  an  indefinite  number 
of  answers  be  obtained.  But  it  is  best  to  compare  those  gains  and 
losses  together  that  have  the  greatest  common  factors;  for  in  such  com- 
parisons, whatever  factors  are  common  can  be  disregarded,  and  the 
remaining  factors  of  each  gain  or  loss  will  show  the  required  quantity 


ALLIGATION  ALTERNATE.  303 

of  the  other  artfcle.     From  the  above  operations  we  derive  the  fol- 
lowing 

Rule.  To  find  whtxt  quantities  of  different  values  shall  be 
taken  to  make  a  mixture  of  a  given  value :  Write  the  different 
values  in  a  column  with  the  medium  value  at  the  left.  Compare 
each  given  value  with  the  value  of  the  mixture.  Write  what  it 
requires  to  equal  that  of  the  mixture  in  a  column  at  the  right  with 
the  sign  -\-  prefixed,  or  what  it  exceeds  that  of  the  mixture  with 
the  sign  — .  Take  such  quantities  of  each  ingredient  that  the 
gains  and  losses  shall  he  equal. 

4:80.  Proof.  Examples  in  Alligation  Alternate  may  be 
proved  by  finding  the  mean  value  of  the  several  ingredients  as 
given  in  the  answer,  and  comparing  it  with  the  given  mean 
value. 

481.  The  following  simple  method  of  solving  this  class  of 
examples  is  sometimes  given,  which  is  preferable  whenever  it 
does  not  give  fractional  portions  of  the  given  quantities. 

III.  Ex.  Let  it  be  required  to  mix  sugar  at  8,  9,  11,  13,  and 
15  cents  per  lb.,  that  the  mixture  may  be  worth  10  cents  per  lb. 

8-f2    Gain  on  lib.  =2    * 

This  deficiency  of  6  in  the 
10  ^  11  —  1    Loss  on  1  lb.  r=      1        gain  may  be  made  up  either  by 

taking  G  lbs.  more  of  the  9-cent 
sugar,  or  3  more  of  the  8-cent, 
or  2  more  of  each  of  the  8  and 
Sum  of  losses  =     9        9-cent  sugars. 
Deficiency  of  gain  =  6 

First  answer,  7  lbs.  at  9  cents,  and  1  lb.  of  each  of  the  others. 
Second  ansiver,  4  lbs.  at  8  cents,  and  1  lb.  of  each  of  the  others. 
Third  answer,  3  lbs.  at  8  and  9  cents,  and  1  lb.  of  each  of  the  others. 

48^.     Examples. 

1.  How  shall  corn  at  50  cents  a  bushel,  be  mixed  with  grain 
at  80  cents  a  bushel,  that  the  mixture  may  be  worth  75  cents. per 
bushel.?  Ans.  1  bu.  at  $.50  to  5  bu.  at  $.80. 


9 

+  1 

<( 

(( 

lib. 

=  1 

11 

—  1 

Loss 

on 

lib. 

— 

1 

13 

—  3 

(( 

(( 

lib. 

— 

3 

15 

—  5 

a 

(( 

lib. 

= 

5 

Sum  0 

f  gains 

—  3 

§04  ALLIGATION.  , 

2.  How  shall  oil  at  80,  95,  and  |1.50  per  gallon,  be  propor- 
tioned that  the  mixture  may  be  worth  $1.00  per  gallon? 

3.  How  shall  tea  at  62,  75,  68,  90,  and  98  cents,  be  propor- 
tioned that  the  mixture  may  be  worth  80  cents  per  lb.  ? 

4.  A  grocer  makes  a  mixture  of  syrup,  worth  62  cents  per 
gall.,  from  syrups  worth  45,  60,  75,  and  80  cents  per  gall. ;  how 
many  gallons  of  each  may  he  use  ? 

5.  A  grocer  has  cider  at  28  and  30  cents  per  gall.,  which  he 
wishes  to  mix  with  vinegar  at  27  cents  per  gall.,  and  water,  so 
that  the  mixture  may  be  worth  25  cents  per  gall. ;  what  propor- 
tions may  he  use  ? 

Ans.  1  gal.  of  each  of  the  other  ingredients  to  f  gal.  water,  etc. 

483.     When  one  of  the  quantities  is  limited,  Jind  the  entire 

gain  or  loss  on  that  quantity,  and  take  such  quantities  of  the  other 

ingredients  that  their  gains  and  losses  shall  balance  each  other  and 

the  gain  or  loss  on  the  limited  quantity. 

When  more  than  one  quantity  is  limited,  Jind  the  resulting  loss 
or  gain  from  taking  the  limited  quantities,  and  balance  as  before. 

III.  Ex.  How  much  tea  at  60,  75,  and  87  cents  per  lb.,  may 
be  mixed  with  30  lbs.  of  tea  at  95  cents  per  lb.,  that  the  mixture 
.shall  be  worth  85  cents  per  lb.  ? " 

Operation. 
60  +  25  X  12  =  -f  300 
75-l-lO-j       1 
87—   2^      5 
1^95  —  10  X  30  =  — 300 
Ans.  12  lbs.  at  60  cents,  1  at  75  cents,  and  5  at  87  cents. 

Examples. 

6.  Plow  many  lbs.  split  peas  at  5  cents  per  lb.,  must  be  put 
with  40  lbs.  coffee  at  21  cents  per  lb.,  that  the  mixture  shall  be 
worth  14  cents  per  lb.?  Ans.  31 1  lbs. 

7.  A  goldsmith  has  gold  16  carats  fine,  which  he  wishes  to 
mix  with  4  oz.  gold  17  carats  fine,  5  oz.  20  carats  fine,  2  oz.  22 
carats  fine,  and  3  oz.  24  carats  fine,  that  the  mixture  may  be  18 
carat!  fine ;  how  many  oz.  of  it  shall  he  use  ?  -t 

n) 


85 


ARITHMETICAL  PROGRESSION.  305 

Note,  —The  term  carat  is  a  word  used  in  indicating  the  proportion  of 
pmre  gold  in  any  given  quantity  of  the  metal ;  thus,  if  the  metal  be  pure 
gold,  it  is  said  to  be  24  carats  fine;  if  two  thirds  gold,  16  carats  fine  ;  if 
17  parts  gold  and  7  parts  alloy,  17  carats  fine,  etc. 

8.  How  much  wool,  of  equal  quantities,  at  35  and  40  cents  per 
lb.,  must  be  mixed  with  100  lbs.  at  60  cents  per  lb.,  that  the  mix- 
ture may  be  worth  45  cents  per  lb.  ? 

4:84:«  When  the  entire  quantity  is  limited,  ^w^  the  proportion 
of  the  ingredients  as  before,  and  then  divide  the  given  quantity 
among  the  ingredients  in  the  proportion  found. 

Examples. 

9.  J.  Blake  has  an  order  from  New  York  for  1000  bushels  of 
wheat,  at  $1.25  per  bushel.  How  shall  he  mix  his  wheat,  which 
he  values  at  $1.20,  $1.22,  and  $1.30,  to  fill  the  order? 

Ans,  100  bu.  at  $1.20,  500  bu.  at  $1.22,  400  bu.  at  $1.30. 

'10.  J.  Smith  wishes  to  purchase  a  farm  of  200  acres,  at  $100      '^ 
an  acre.     How  much  woodland  at  $125  per  acre,  mowing  up-     ^ 
land  at  $90  per  acre,  pasture  land  at  $70  per  ^cre,  and  tillage 
ground  at  $128  per  acre,  may  he  purchase?)  ^  w  ->/'  * '    L\(J  ^h 

11.  How  many  lbs.  of  cotton  at  60,  73,  and  98  cents  per  lb., 
must  be  mixed  with  750  lbs.  at  90  cents,  that  the  mixture  may 
contain  2000  lbs.  at  80  cents  per  lb. ?.  A/O  Vr /i     A  ^Omy^'^. 

Note.  — First  balance  the  loss  on  the  750  lbs.  with  gaiit  on  one  of  the-^^  > 
other  ingredients  taken;  then  proceed  to  make  a  mixture  of  the  other ^i  . 
ingredients  equal  to  the  entire  quantity  given,  minus  the  quantities  bal,  N  V 
ajiced.  '     I 


ARITHMETICAL   PROGRESSION. 

485.  Arithmetical  iProgression  is  progression  by  equal 
differences. 

486.  An  Arithmetical  Series  is  a  succession  of  numbers 
which  increase  or  decrease  by  a  common  difference. 

If  the  numbers  increase  from  the  first  term,  the  series  i«  an 
Increasing  Series:  e.  g.,  2,  4,  6,  8,  10,  12,  &c. 
20 


306  ARITHMETICAL  PROGRESSION. 

If  the  numbers  decrease  from  the  first  term,  the  series  i?  a 
Decreasing  Series;  e.  g.,  13,  11,  9,  7,  5,  &c. 

^rST,  In  every  series,  five  tilings  are  to  be  considered;  viz., 
the  First  Term,  the  Last  Term,  the  Namher  of  Terms,  the 
Common  Difference,  and  the  Sum  of  the  Terms ;  any  three  of 
which  being  given,  the  other  two  may  be  found.  Tiiis  gives  rise 
to  twenty  distinct  cases,  a  few  of  the  more  important  of  which 
will  be  here  presented. 

Note  I.  —  For  the  remaining  cases,  also  for  full  discussions  of  Geo- 
metrical Progression  and  Annuities,  the  student  is  referred  to  works  on 
Algebra. 

Note  IT.  — Increasing  series  only  will  be  considered  in  this  book,  as 
rules  that  apply  to  increasing  series  apply  to  decreasing  series  also,  pro- 
vided that,  wherever  the  common  difference  is  introduced,  it  is  used  with 
the  contrary  sign. 

488.  To  FIND  ANY  Term  in  a  Series,  when  the  First 
Term,  Common  Difference,  and  Number  of  Terms 
are  given. 

Let  5  =  first  term,  2  =  common  difference,  and  6z=the  number  of 
terms.     The  series  will  be  constructed  as  follows :  — 

(1.) 
Ist  term.     2d  term.     3d  term.       4th  term.       5th  term.       6th  term. 
5.  5  +  2.     5  +  2X2.     5  +  3X2.     5+4X2.     5  +  5X2. 

We  find  that  the  second  term  equals  the  first  term,  plus  the  common  . 
difference ;  the  third  term  equals  the  first  term,  plus  two  times  the  com- 
mon difference ;  the  fourth  term  equals  the  first  term,  plus  three  times 
the  common  difference,  &c. ;  and  that  the  last  or  sixth  term  equals  the 
first  term,  plus  five  times  the  common  difference.     Hence, 

I.  To  find  any  term  of  the  series :  Add  the  first  term  to  the 
'product  of  the  common  difference  multiplied  hy  the  number  of 
terms  which  precede  it. 

II.  To  find  the  last  term:  Add  the  first  term  to  the  product  of 
the  common  difference  multiplied  hy  the  number  of  terms  less  one. 


ARITHMETICAL  PROGRESSION.  307 

Examples. 

1.  In  an  increasing  series  the  first  term  is  4,  and  the  common 
difference  is  8  ;  what  is  the  seventh  term  ?  Atis.  52. 

2.  The  first  term  is  7,  the  common  difference  ^,  and  the.4iura- 
ber  of  terms  20 ;  what  is  the  last  term  ?  /  5  T 

3.  If  5  lbs.  of  power  is  imparted  to  a  fly-wheel  at  eachl-evolu- 
tion,  what  is  its  power  at  the  end  of  the  tenth  revolution  from  a         ' 
state  of  rest,  provided  its  average  loss  of  power  from  friction  an^    ,, 
other  causes  is  1  lb.  during  each  revolution  ?  Ans.  40  lbs.    ""  r 

4.  If  a  stone,  in  falling  to  the  earth,  descends  IGyi^-  feet  during 
the  first  second,  3  X  16^^  feet  during  the  next,  5  X  16iV  feet 
during  the  third,  and  so  on ;  how  far  will  it  fall  during  the  elev* 
enth  second  ?    j  >  )   f- 

5.  What  is  the  amotmt  of  $200  at  pimple  interest  for  8  years, 
at  6  per  cent.  ?    ^  f  b       ^  "  '  •'   '-'  ■  ^^"^    ■  H  ^  r    •  '     . 

Note.  —  The  amount  will  be  the  ninth  term  of  the  series,  of  which  the 
first  term  is  $200. 

4:89.     To  FIND  THE  Common  Difference  in  a  Series, 
ALSO  THE  Number  of  Terms. 

If,  in  series  (1.)  we  subtract  the  first  term  from  the  last,  we  have  r8o 
maining  5X2,  that  is,  the  common  difference  multiplied  by  the  number 
of  terms  less  one.    Hence, 

I.  To  find  the  common  difference :  Divide  the  difference  be- 
tween the  first  and  last  term  by  the  number  of  terms  less  one. 

II.  To  find  the  number  of  terms :  Divide  the  difference  between 
the  first  and  last  term  by  the  common  difference,  and  add  one  to 
the  quotient. 

Examples. 

6.  The  first  term  of  a  series  is  7,  the  last  term  19.  and  the 
number  of  terms  13 ;  what  is  the  common  difference  ?       Ans.  1. 

7.  The  first  term  is  30,  the  last  term  is  3,  and  the  number  of 
terms  10  ;  what  is  the  common  difference  ?   J 

8.  The  first  term  is  8,  the  last  term  23,  and  the  common  dif- 
fe>^nce  1^ ;  required  the  number  of  terms,  f  | 


308  ARITHMETICAL  PROGRESSSION. 

9  A  boy,  in  picking  up  stones  2  feet  apart,  and  carrying  them, 
one  at  a  time,  to  a  deposit  2  feet  from  the  first,  found  that  to 
carry  the  last  one,  he  had  walked  60  feet ;  how  many  stones  did 
he  carry  in  all?  Ans.  15  stones. 

490.     To  FIND  THE  Sum  of  the  Series. 
Let  2,  4,  6,  8,  10,  12,  14,  16,  be  a  series,  of  which  we  wish  to  find 
the  sum.    We  write  under  it  the  same  series  in  an  inverted  order,  an^ 
add  the  terms  as  follows :  — 


2 

4 

6 

8 

10 

12 

14 

16 

16 

14 

12 

10 

8 

6 

4 

2 

18        18         18         18         18         18         18         18 
We  then  have  the  sum  of  both  series  z=  8  X  18, 
or  the  sum  of  one  series  in:  s.J:yiK 

But  8  equals  the  number  of  terms,  and  18  the  sum  of  the  extremes. 
Hence, 

To  find  the  sum  of  a  series  :  Multiply  one  half  the  sum  of  the 
extremes  hy  the  number  of  terms. 

Examples. 

10.  The  first  term  of  a  series  is  4,  the  last  40,  and  the  number 
of  terms  11 ;  what  is  the  sum  of  the  series  ?  Ans*  242. 

11.  What  is  the  sum  of  the  odd  numbers  from  1  to  99  inclu- 
sive? /i   .  ]  ^(r6' 

12.  What  is  the  sum  of  the  multiples  of  3  from  6  to  45  in- 
elusive  ? 

13.  How  many  notes  must  a  person  sing  in  ascending  two 
octaves,  if  he  goes  back  to  the  first  note  each  time  he  strikes  rt 
new  one,  and  sounds  all  the  intermediate  notes  each  time  he 
ascends?  Ans.  120  notes. 

14.  Two  of  Dio  Lewis's  pupils  tried  their  skill  in  running  for 
pegs.  Each  set  up  5  pegs  6  feet  apart,  and  commenced  running 
6  feet  from  the  first  peg.  How  far  did  each  run  to  place  the 
pegs  at  his  starting-point  ?   /  ^    '" 

15.  How  far  would  the  first  boy  of  a  row  of  21  scholars  travel, 
Jn  gathering  writing-books  from  the  row,  if  the  scholars  were  2^ 
feet  apart,  and  he  brought  one  book  at  a  time  to  his  own  desk  ?        . 


GEOMETRICAL  PROGRESSION.  309 


GEOMETRICAL    PROGRESSION. 

4:91.  Geometrical  Progression  is  progression  by  equal 
multipliers. 

4:9^.  A  Geometrical  Series  is  a  succession  of  numbers 
which  increase  or  decrease  by  a  common  multiplier.     Thus, 

2,  4,  8,  16,  32,  64,  is  an  increasing  geometrical  series,  in  which 
the  multiplier  is  2. 

2)  Ij  h  h  h  tV'  ^s  ^  decreasing  geometrical  series,  in  which 
the  multiplier  is  ^. 

4.93.    The  common  multiplier  is  called  the  Katio. 

494.  In  every  geometrical  progression,  five  things  are  to  be 
considered ;  viz.,  the  First  Term,  the  Last  Term,  the  Number  of 
Terms,  the  Common  Eatio,  and  the  Sum  of  the  Terms  ;  any  three 
of  which  being  given,  the  other  two  may  be  found. 

4:9«>.    To  FIND  THE  Last  Term  of  a  Seriks,  the  First 
Term,  the  Ratio,  and  Number  of  Terms  being  given. 

Let  3  be  the  first  term,  2  the  ratio,  and  5  the  number  of  terms.  The 
series  will  then  become, 


(1.) 

1st  term. 

2d  term. 

3d  term. 

4th  term. 

5th  term. 

3, 

3X2, 

3X2^ 

3  X  2^ 

3X2^ 

in  which  the  second  terra  equals  the  first  term  multiplied  by  the  ratio, 
the  third  term  equals  the  first  term  multiplied  by  the  second  power  of 
the  ratio,  the  fourth  term  equals  the  first  term  multiplied  by  the  third 
power  of  the  ratio,  and  the  fifth  term  equals  the  first  term  multiplied 
by  the  fourth  power  of  the  ratio.     Hence, 

I.  To  find  any  term  of  the  series :  Multiply  the  first  term  hy 
the  ratio  raised  to  a  power  equal  to  the  number  of  terms  which 
precede  the  required  term. 

II.  To  find  the  last  term  of  the  series  :  Multiply  the  first  term 
hy  the  ratio  raised  to  a  poioer  equal  to  the  number  of  terms  less 


SXO  GEOMETRICAL  PROGKESSION. 

Examples. 
/^        1.  What  is  the  seventh  term  of  the  series  2,  6,  18,  54,  &c.?/  L^f 
■ — T^lT^'  ^^^^  ^^  *^^^  fifteenth  terra  of  the  series  5,  2^,  1  J,  |,  -j^^^,  &c.  ? 
r-^  ^'   3.  What  is  the  amount  of  $500  for  7  years,  at  6  per  cent., 
compound  interest  ? 

Note. —  1.06  is  the  ratio,  and  the  amount  the  eighth  term  of  the  series. 

—  4.  Naturalists  have  found  that  the  ratio  of  increase  of  some 
kinds  of  animalcute  (microscopic  animals)  is  often  four  in  a  single 
day.  At  that  rate,  what  would  be  the  increase  of  one  animal- 
cula  and  its  descendants  in  ten  days?  Ans.  1,048,576. 

4:96*     To  FIND  THE  Ratio,  the   First  Term,  the  Last 
Term,  and  Number  of  Terms  being  given. 

In  series  (1),  if  the  last  term,  3  X  2^,  be  divided  by  the  first  term,  3, 
the  quotient  will  be  2\  or  the  fourth  power  of  the  ratio,  the  fourth  root 
of  which  will  equal  the  ratio.     Hence, 

To  find  the  ratio :  Divide  the  last  term  hy  the  first  term,  and 
extract  that  root  of  the  quotient  whose  index  equals  the  number  of 
terms  less  one. 

Examples. 

5.  The  first  term  of  a  series  is  2,  the  last  term  128,  the  num- 
ber of  terms  3  ;  what  is  the  ratio  ?  Ans.  8. 

6.  The  first  term  is  4,  the  last  term  J,  and  the  number  of  terms 
4 ;  what  is  the  ratio  ?  Ans.  ^. 

7.  The  extremes  are  5  and  625,  and  the  numbt^r  of  terms  4 ; 
what  is  the  ratio?  Oii, ,  * 

^;49T,     To  FIND  the  Sum  of  a  Series. 

Let  3,  9,  27,  81,  243,  be  a  series,'  of  which  we  wish  to  find  the 
sum. 

OPERxiTIOX. 

3  times  the  first  series  =  9     27     81     243     729 ;  subtracting 

the  first  series  =i      3     9     27     81     243,  we  have 


twice  the  first  series  =  — 3     0      0       0        0     729,  or  729  ■— 3. 
.-.  the  first  series  z::^  i^^Tj^^^. 


ANNUITIES.  314 

By  multiplying  each  term  of  the  series  by  the  ratio,  3,  we  have  a 
second  series,  \vhose  sum  is  3  times  that  of  the  first  series,  from  which 
we  subtract  the  first  series ;  the  remainder  equals  twice  the  sum  of  the 
first  series,  wliich  we  find  by  dividing  by  2.     Hence, 

To  find  the  sum  of  the  series  :  Subtract  the  first  term  from  the 
product  of  the  last  term  multiplied  by  the  ratio ;  divide  the  remain- 
der by  the  ratio  less  one. 

Note.  —  If  the  series  is  descending,  the  last  term  multiplied  by  the 
ratio  should  be  taken  from  the  first  term,  and  the  remainder  be  divided 
by  one  less  the  ratio. 

Examples. 

8.  What  is  the  sum  of  the  series  3,  12,  48,  192,  768,  3072? 

Ans.  4095. 

9.  The  first  term  is  5,  the  last  term  3125,  and  the  number  of 
terms  5  ;  what  is  the  sum  of  the  series  ?  Ans.  3905. 

10.  What  is  the  sum  of  7  terms  of  the  series  4,  8,  16,  32,  &c.  ? 

11.  If  1  of  the  air  in  a  receiver  be  taken  from  it  by  an  air- 
pump  at  the  first  stroke  of  the  piston,  and  ^  of  the  remainder  at 
the  second  stroke,  and  so  on,  what  will  be  the  amount  taken 
from  the  receiver  by  8  strokes  ?  Ans.  flf. 


ANNUITIES. 


408.  Annuities  are  periodical  payments  of  fixed  sums  of 
money,  in  consideration  of  money  paid  or  services  rendered. 

409.  When  an  annuity  is  made  for  a  definite  number  of 
years,  it  is  called  a  certain  annuity ;  when  it  is  made  forever,  a 
perpetuity ;  when  it  depends  upon  the  life  of  one  or  more  per- 
sons, a  life  annuity  ;  when  it  does  not  commence  till  a  given  time 
has  elapsed,  it  is  said  to  be  in  reversion. 

^00.  When  annuities  are  granted  by  government,  they  are 
called  Pensions. 

0O1.  The  Amount  of  an  annuity  is  the  sum  of  ^11  the  pay 
ments,  plus  their  interest,  from  the  time  they  become  due. 


312  ANNUITIES. 

«>0d.  The  Present  Worth  of  an  annuity  is  such  a  sum  of 
money  as,  put  at  interest,  will  exactly  pay  the  annuity. 

^03.  Annuities  are  said  to  be  in  Arrears,  or  Foreborne, 
when  they  remain  unpaid  after  they  become  due. 

504:,  Annuities  are  generally  computed  at  compound  inter- 
est. 

ANNUITIES  AT  SIMPLE  INTEREST. 

«>0^.  III.  Ex.  What  is  the  amount  of  an  annuity  of  $200 
a  year,  at  6  per  cent,  simple  interest,  5  years  in  arrears  ? 

The  payment  due  at  the  end  of  the  fifth  year  is  $200  ,•  that  which 
was  due  at  the  end  of  the  fourth  year  amounts,  at  the  end  of  the  fifth 
year,  to  $200  plus  the  interest  on  the  same  for  1  year;  that  which  was 
due  at  the  end  of  the  third  year,  to  $200  plus  its  interest  for  2  years ; 
that  due  at  the  end  of  the  second  year,  to  $200  plus  its  interest  for  3 
years  ;  that  due  at  the  end  of  the  first  year,  to  $200  plus  its  interest 
for  4  years.  Hence,  the  sums  due  at  the  end  of  the  fifth  year  would 
form  an  arithmetical  series,  200,  200  +  12,  200  +  24,  200  +  36,  200  + 
48,  of  which  the  first  term  is  $200,  the  last  term  the  amount  of  $200 
for  the  number  of  years  less  1,  and  the  number  of  terms  the  number 
of  years.     Hence  the  sum  may  be  found  by  Art.  490 ;  and,  generally, 

To  find  the  amount  of  an  annuity  at  simple  interest :  Find  the 
sum  of  an  arithmetical  series,  of  which  the  first  term  is  the  last 
payment,  the  last  term  the  amount  of  the  frst  payment,  and  the 
number  of  terms  the  number  of  payments. 

Examples. 

1.  What  is  the  amount  of  an  annuity  of  $300  for  6  years,  at 

6  per  cent.,  simple  interest  ?  Ans,  $2070. 

2.  What  is  the  amount  of  an  annuity  of  |600  for  7  years,  at 

7  per  cent,  simple  interest  ? 

3.  A  gentleman's  salary  of  |?12O0  a  year,  payable  quarterly, 
remained  unpaid  for  4  years ;  what  was  then  his  due  ? 

ANNUITIES  AT  COMPOUND  INTEREST. 
^06«    III.  Ex.   What  is  the  amount  of  an  annuity  of  |36 
for  4  years,  at  6  per  cent.,  compound  interest  ? 


ANNUITIES  AT  COMPOUND  INTEREST. 


313 


We  will  first  find  the  amount  of  an  annuity  of  $1  for  the  same  time. 

The  last  payment,  due  at  the  end  of  4  years,  will  be  $1.  The  sum 
due  on  the  third  payment,  at  the  end  of  the  fourth  year,  will  be  the 
amount  of  $1  for  1  year;  that  due  on  the  second  payment  will  be  the 
amount  of  $1  for  2  years;  that  due  on  the  first  payment  will  be  the 
amount  of  $1  for  3  yeai's.  Hence  the  four  sums  due  will  form  the 
geometrical  series, 

1,     1.06,         1.1236,  1.19101,  or 

1,     1.06,     1  X  ^1.06)2     1  X  (1.06)3, 
of  which  the  first  term  is  the  last  payment,  the  last  term  the  amount 
of  the  first  payment,  and  the  number  of  terms  the  number  of  payments. 
Finding  the  sum  of  this  series  (Art.  297),  and  multiplying  by  36,  we 
obtain  the  required  amount.     Hence,  the 

Rule.  To  find  the  ^amount  of  an  annuity  at  compound  inter- 
est :  Find  the  anwunt  of  an  annuity  of  $1  for  the  given  time  by 
geometrical  progression  (Art.  497),  and  multiply  the  sum  thus 
<ihtained  by  the  annuity. 

Examples. 

^44/ What  is  the  amount  of  an  annuity  of  $1  for  2  years,  at  6 
per  cent. ?  for  3  years ?  for  5  years?  for  10  years? 

2.  What  is  the  amount  of  an  annuity  of  $20  for  8  years,  at  5 
percent.?  ^ws.  #190.98. 

507.     Table  L, 

Showing  the  amount  of%l  or  £1  annuity  from  1  year  to  20. 


Years. 

5  Per  Cent. 

6  Per  Cent. 

Years. 

5  Per  Cent. 

6  Per  Cent. 

1 

1.000000 

1.000000 

11 

14.206787 

14.971643 

2 

2.050000 

2.060000 

12 

15.917127 

16.869941 

3 

3.152500 

3.183600 

13 

17.712983 

18.882138 

4 

4.310125 

4.374616 

14 

19.598632 

21.015066 

5 

5.525631 

5.637093 

15 

21.578564 

23.275970 

6 

6.801913 

6.975319 

16 

23.657492 

25.672528 

7 

8.142008 

8.393838 

17 

25.840366 

28.212880 

8 

9.549109 

9.897468 

18 

28.132385 

30.905653 

9 

11.026564 

11.491316 

19 

30.539004 

33.759992 

10 

12.577893 

13.180795 

20 

33.065954 

36.185591 

Note.  —  The  following  examples  may  be  performed  by  the  use  of 
Table  I. 


314 


ANNUITIES. 


I  l^hat  are  the  amounts  of  the  following  annuities  ? 

3.  $100  for  7  years,  at  5  per  cent. 

4.  $200  for  10  years,  at  6  per  cent. 
#  5.  £150  for  18  years,  at  6  per  cent.  ^C.  -  v   /;,<,    ■. 
V    6.  A  gentleman,  on  his  daughter's  first  birthday,  and  on  each 

succeeding  birthday,  deposited  $10  in  a  savings-bank,  which 
yielded  5  per  cent,  compound  interest,  and  presented  her  with 
the  amount  on  her  eighteenth  birthday.  What  was  the  value  of 
the  present  ?     ^  fr  /,  j  <^  j  -f  ^^*i4 

508.  To  find  the  present  worth  of  an  annuity  :  Divide  the- 
amount  of  the  annuity  by  the  amount  of  %1  compound  interest  for 
the  time  given.  It  may  also  be  obtained  by  the  use  of  the  fol- 
lowing table :  — 

Table  II., 

Showing  the  present  value  of  an  annuity  of  t^  or  £1  from 
1  year  to  20. 


Years. 

5  Per  Cent. 

6  Per  Cent. 

Years. 

5  Per  Cent. 

6  Per  Cent. 

1 

0.952381 

0.943396 

11 

8.306414 

7.886875 

2 

1.859410 

1.833393 

12 

8.863252 

8.383844 

3 

2.723248 

2.673012 

13 

9.393573 

8.852683 

4 

3.545950 

3.465106 

14 

9.898641 

9.294984 

5 

4.329477 

4.212364 

15 

10.379658 

9.712249 

6 

5.075692 

4.917324 

16 

10.837770 

10.105895 

7 

5.786373 

5.582381 

17 

11.274066 

10.477260 

8 

6.463213 

6.209794 

18 

11.689587 

10.827603 

9 

7.107822 

6.801692 

19 

12.085321 

11.158116 

10 

7.721735 

7.360087 

20 

12.462210 

11.469421 

7.  What  is  the  present  worth  of  an  annuity  of  $200  for  4 
years,  at  5  per  cent.?  Ans.  $709.19. 

8.  What  must  I  pay  for  an  annuity  of  8300  for  10  years,  at  6 


per  cent.i 


5©9,     Questions   for   Eeview. 


What  is  Alligation  Medial  ?    Al' 


Of  what  does  Alligation  treat  ? 
ligation  Alternate  ?     What  other  name  might  be  used  for  Alligation  ? 
Make  an  example  in  Alligation  Medial ;  perform  and  explain  it,  and 


QUESTIONS  FOJl  REVIEW.  315 

grm  the  rule.  Give  the  proof.  Make  an  example  in  Alligation  Alter- 
nate ;  perform  and  explain  it,  and  give  the  rule.  Give  the  proof.  How 
many  answers  may  you  have  to  examples  in  Alligation  Alternate  ? 
How  do  you  proceed  when  one  quantity  is  limited  ?  when  several  quan- 
tities are  limited?  when  the  entire  quantity  is  limited? 
'"'^What  is  Arithmetical  Progression  ?  What  is  an  Arithmetical 
Series  ?  When  is  a  series  increasing  ?  when  decreasing  ?  Give  exam- 
ples of  each.  How  many  things  are  to  be  considered  in  a  series  ?  How 
many  must  be  known,  that  the  rest  may  be  found  ?  To  what  is  the  last 
term  of  a  series  equal  ?  Show  why  ?  How  do  you  find  the  common 
"difference  ?  Explain.  How  the  number  of  terms  ?  Explain.  How  the 
sum  of  the  series  ?     Explain. 

^_What  is  Geometrical  Progression  ?  a  Geometrical  Series  ?  What 
is  the  constant  multiplier  called  ?  When  the  ratio  is  greater  than  uni- 
ty will  the  series  be  increasing  or  decreasing  ?  What  will  it  be  when 
the  ratio  is  less  than  one  ?  How  many  things  are  to  be  considered  in 
a  Geometrical  Series  ?  How  many  must  be  known,  that  the  rest  may 
be  found  ?  What  is  your  rule  for  finding  the  last  term  ?  explain  it. 
Your  rule  for  finding  the  ratio  ?  explain  it.  Your  rule  for  finding  the 
sum  ?  explain  it. 

What  are  Annuities  ?  What  is  a  Certain  Annuity  ?  a  Perpetuity  ? 
a  Life  Annuity?  a  Pension?  What  is  the  amount  of  an  annuity?  the 
present  worth  ?  W^^'^  ^^'^  annuities  in  arrears  ?  How  are  they  gen^ 
erally  computed  ?  Give  your  rule  for  finding  the  amount  of  an  annui- 
ty at  simple  interest.  Illustrate  it  by  an  example.  Give  your  rule  for 
finding  the  amount  of  an  annuity  at  compound  interest.  Illustrate  it 
by  an  example.  How  do  you  use  the  table  ?  How  do  you  find  the 
present  worth  of  an  annuity  ? 


510,    Miscellaneous  Examples. 

1 1  _' 

1.  ^  of  a  number  exceeds  |  by  20  ;  what  is  that  number?  ^ 

21 

2.  The  sum  of  two  numbers  being  4^,  and  one  of  them  being 

45-1-         244 
the  difference  betw^een  -j^-  and  -~,  what  is  the  other  ? 

3.  From  the  product  of  the  sum  and  difference  of  3.6  and  2.24, 
take  the  difference  between  the  squares  of  3.6  and  2.24.   £) 


316  MISCELLANEOUS   EXAMPLES. 

5.  A  can  mow  2  acres  in  a  day,  B  2^  acres,  and  C  3  acres*, 
what  is  the  smallest  number  of  acres  that  will  give  a  number  of 
whole  days' work  for  either  ?.J//^     f-  :  r 

6.  What  is  the  longest  rail  that  will  exactly  fence  either  side 
of  a  lot  of  ground,  the  sides  being  bQ  ft.,  42  ft.,  63  ft.,  77  ft.,  and 

49  ft.?      y 


k^\ 


7.^xJ4^P^8.^|x4.75x83iX27-f^?^/5 

9.  A  quotient  being  95 1,  and  the  divisor  .33^  of  3024/^,  what 
is  the  dividend  ?     fC  ^f  (Jj\ 

10.  Bought  a  piece  of  cloth  for  $75.30,  sold  f  of  it  to  one  per- 
son, and  ^  of  the  remainder  to  another ;  what  is  the  value  of  the 
unsold  part  at  10^  advance  upon  the  cost?    /  ^  1  J^ 

11.  An  aeronaut  ascends  at  the  rate  of  4J-  miles  an  hour  for  40 
minutes,  after  which  he  maintains  the  same  elevation ;  if  his  bal- 
loon is  driven  east  7  miles  during  the  first  hour  from  the  time  of 
his  starting,  and  in  an  opposite  direction  at  the  rate  of  10  miles 
an  hour  for  the  remaining  time,  how  far  from  his  starting-point  in 
a  straight  hne  is  he  at  the  end  of  5  hours  ? J  y*  /  f 

"12.  What  is  the  weight  of  a  bale  of  cotton  cloth,  containing  13  .     i 
pieces,  42  yards  to  the  piece,  every  3  yards  weighing  1^  lbs.  ?     ^^  j  "o" 

~13.  A  trader  bought  apples  at  $1.62^  per  barrel,  and  immedi- 
ately sold  them  at  $2.25,  making  $234.37^ ;  how  many  barrels 
were  bought  ? 

-  14.  Divide  380  -[-  20  +  5  -f   81  —  69    X  5  by  7.5  —  .5  -|-  . 

9.9-^3—1.8 

.1  *  ■" 

15.  The  sum  of  three  numbers  is  55^ ;  two  of  them  are  14|  and 
24} i;  what  is  the  third?  1 

16.  Suppose  a  dividend  to  be  241.3,  and  the  quotient  .127 ; 
what  was  the  divisor  ? 

17.  When  the  ice  upon  a  pond  is  10  inches  thick,  what  will 
be  the  value  of  the  ice  taken  from  one  acre  of  the^  pond  at  ^-  ct. 
per  lb.,  1  cu.  ft.  containing  58 1  lbs.?  /,        s  ^ 

18.  A  can  do  ^  of  a  piece  of  work  in  4  days,  B  ^  of  it  in  5 


I- 


MISCELLANEOUS  EXAMPLES.  317 

days,  and  C  J  of  it  in  2  days ;  in  what  time  will  all  together  do 
the  whole  ?  .    ^  J^- 

~19.  ^  +  ^  +  i  -|-  If  of  a  certain  number  increased  by  3^  -[" 
10|  =  50f  ;  what  is  the  number?  ^.^  :--^^ 

20.  A  man  left  ^  of  his  property,  waiting  $2000,  to  his  daugh- 
ter, I  of  it  to  his  son,  and  the  balance  to  his  widow,  whose  share 
was  $500  more  than  that  of  his  daughter ;  what  was  the  share  of 
each?     Ans.  Widow's,  $7250;  daughter's.  $6750;  son's  $3500. 

21.  Of  what  number  is  ?  the  I  part?     -%     ,        ) 

22.  A  can  do  a  piece  of  work  in  1^  hours,  A  and  B  in  48  min. ; 
in  what  time  can  B  do  it  alone  ? 

23.  A  cubic  inch  of  infusorial  earth  in  Germany,  weighing  220 
grains,  was  found  to  consist  of  41  billions  of  infusoria ;  what  was 
the  w%ight  of  one  skeleton ?  {     r^, , . ,, ,.  .....^    yl     ^  ^^  ^tft'i 

24.  ^  Oambi^idg^,'  Sept.  1,  1864  ^ 
Mr.  James  H.  Eaton, 

To  Frank  Caldwell,  Dr, 

To  40000  ft.  pine  boards, (a)  $12.00  per  M. 

"    15625  "  walnut, to)    15.00    "     "j 

"     2875  "scantling, (a)      3.75    "     « 

"    23^  thousand  shingles, ^      4.50    "     " 

"  23  planks,  12  ft.  long,  11  in.  by  3  in.,*  .  ,  (a)  12.50  "  " 
«  12stickstimber,  29ft.long,  10in.byl2in.,  ^  13.62  «  « 
Required  the  cost  of  the  above  ?  A^is.  $887.79 

25.  A  merchant  buys  gloves  at  3s.,  and  sells  them  at  4s.  6d. ; 
what  does  he  gain  in  laying  out  50  £?     ,  .  4 (5'"^ 

26.  Divide  52  into  two  such  parts  that  -i  of  one  part  shall 
equal  §  of  the  other.  A7is.  12  and  40. 

27.  A  person  being  asked  the  time  of  day,  answered  that  it 
was  between  two  and  three  o'clock,  and  that  f  of  the  time  past     . 
two  equalled  |  of  what  it  wanted  of  three.     Required  the  time.    /  -r- 

28.  There  is  a  pole  18  ft.  long,  standing  in  such  a  position  that 
^  of  the  part  in  the  mud  is  equal  to  |  of  the  part  in  the  water, 
and  to  y\  of  the  part  in  the  air  ;  how  many  feet  are  there  in  each 
element  ? 

Ans.  3  ft.  in  the  mud,  ^  ft.  in  the  water,  and  10 J-  ft.  in  the  air. 

*  See  Note,  page  141. 


S18  MISCELLANEOUS  EXAMPLES. 

29.  Suppose  the  water  to  rise  so  that  -^  of  the  part  in  the  mud 
is  equal  to  ^  of  the  part  in  the  water,  how  many  feet  are  there  in 
each  element?  ^ 

Ans.  3  ft.  in  the  mud,  6  ft.  in  the  water,  and  9  ft.  in  the  air,     i  ^ 

30.  Butter  being  worth  25  cents  per  lb.,  if  f  of  a  lb.  will  pay 

for  f  of  a  dozen  eggs,  how  many  eggs  will  be  required  to  pay  fa?     ^ 
6  lbs.  of  raisins,  7  lbs.  of  which  cost  98  cents  ?  /^  d  c         *^^ 

31.  A,  B,  and  C  shared  $102  between  them,  so  that  A  had 
^17  more  than  B,  and  B  had  $20  more  than  C ;  what  had  each?av)'A 

32.  The  captain  of  a  ship  at  sea  finds  by  his  chronometer,  at 
12  o'clock  at  noon,  that  it  is  45  m.  past  8  o'clock  in  the  evening 
at  London ;  what  is  his  longitude  ? 

33.  When  it  is  10  A.  M.,  in  X,  which  is  44°  15'  2"  W.  long., 
what  is  the  time  in  Y,  which  is  8°  4'  40"  E.  long.?       /      y 

34.  When  it  is  half  past  10  o'clock,  P.  M.,  of  J)ec.  31,  1865,  in 
Albany,  N.  Y.,  what  is  the  time  in  Cohstantihopfe?"(Tsl^  Art.^O&f 

i    35.  What  is  the  exact  time  from  Nov.  19,  1858,  18  min.  of  4, 
P.  M.,  to  Apr.  9,  1863,  10  min.  past  2,  P.  M.  ? 

Ans.  4  yrs.  140  d.  22  h.  28  min. 

36.  Paid  6|  cents  a  pound  for  3200  lbs.  of  pork;  on  the  sale 

of  f  of  it  I  gained  $8^  ;  with  the  money  received  I  purchased  7|-        •- 
tons  of  plaster ;  how  many  cords  of  wood,  of  "v^hioh  3  cords  cost 
I22J-,  would  be  worth  1  ton  of  the  plaster  ?   -    r^l" 

37.  If  $1239  were  paid  for  harvesting  the  wheat  on  a  lot  of 
land  400  rods  long,  350  rods  wide,  what  should  be  paid  for  har- 
vesting the  oats  upon  a  lot  500  rods  long,  450  rods  wide,  the  cost 

of  harvesting  oats  being  |  as  much  as  for  harvesting  wheat  ?  '      /  3  ^ 

38.  A  coal  dealer  purchased  500  tons  of  coal  at  17.50  per  long 
ton,  paid  $1  per  ton  for  freighting,  and  sold  it  for  $11  by  the  short 
ton;  what  per  cent,  did  he  gain?  Ans.  $44ff  %. 

39.  When  gold  sells  at  59%  advance,  how  much  can  be  bought 
With  1100  of  good  bank  bills  ? 

40.  A  city  collector  has  .8  %  for  collecting  taxes ;  he  pays  into 
ihe  treasury  $94625.64  after  deducting  his  commission;  what 
was  the  whole  sum  collected? 

41.  An  army  of  50000  men  besieged  a  city  for  three  months; 


MISCELLANEOUS   EXAMPLES.  319 

during  the  first  month,  5%  were  lost  by  sickness  and  desertion; 
at  the  beginning  of  the  second  month  an  accession  was  made  to 
their  force  equal  to  20%  of  what  remained;  during  the  second 
month  they  lost  25  %  of  the  men,  and  during  the  last  month  30% 
of  what  then  remained  were  lost  and  detailed  for  service  else- 
where ;  how  many  were  left?  ■J'  V  S'  j^  ^  " 

42.  A  grocer  imported  75  galls,  of  oil,  which  cost  him  $2  a 
gall,  and  a  duty  of  10%.  Suppose  5  galls,  to  leak  out,  for  what 
must  he  sell  the  remainder  per  gall,  to  gain  10%  on  the  money 
spent  ?  .;  '-      '        '*    ^ 

43.  If  you  buy  figs  at  the  rate  of  9  lbs.  for  $1.50,  and  sell  them 
at  the  rate  of  10  lbs.  for  |2,  what  do  you  gain  per  cent.?^'' 

44.  Pedrick  &  Closson  sold  at  auction, 

2  mattresses,      at  $16.00,  which  cost  $13.50. 


8  chairs,             at 

4.62, 

a 

a 

3.75. 

1  rocker,             at 

17.50, 

a 

a 

17.00. 

1  set  furniture,  at 

38.00, 

iC 

li 

42.00. 

1    '*         "          at 

83.50, 

it 

a 

62.00. 

4^^ 


They  also  sold  on  commission,  at  10%, 

5  chairs,    at  $8.00  1  table,     at  $8.00. 

12      "        at      1.70.         1  lounge,  at  12.00. 
1  bureau,  at    18.00.         1  stove,    at  17.00. 
What  were  their  net  proceeds  from  the  above  sales  ?  -^4 

45    I  found,  on  going  to  Rand  &  Tyler's  dry  goods  store,  that 
they  had  that  morning  marked  up  their  goods  15%  ;  what  did  I 
save  by  purchasing  the  following  goods  the  day  before  ? 
18  yds.  blk  silk,  at    $1.12. 
13     «    de  laine,  at        .27. 
9     "    cambric,  at        .15. 
3     '*    silesia,     at        .25.  ^ 

1  waterproof,     at      8.00.       A\  ^  ^ 

46.  Paid  84  cents  a  gallon  for  a  cask  containing  27  galls,  of 
kerosene,  10%  of  which  leaked  out;  if  the  remainder  was  sold 
25  %  on  the  gallon  higlier  than  it  cost,  what  was  the  gain  or  loss 
on  the  money  invested?    g<i.\.    J^  '  3  ^'^ 

47.  Sold  6  sewing  machines  at  |72  each.     On  two  of  them  I 


S20  MISCELLANEOUS  EXAMPLES. 

gained  20*^,  on  two  others  33 i%,  and  on  the  others  I  lost  25%; 
what  was  the  balance  of  gain  or  loss  ?j 

48.  I  sell  a  lot  of  carrots  at  $13.26  per  ton,  and  take  in  payment 
a  note  for  2  months  without  interest,  gaining  8^%  ;  what  did  I 
pay  per  ton  I' f^  J/  ir 

49.  What  is  the  interest  on  £73  from  Oct.  21,  1858,  to  May 

11,1860?         #^(;,^/?f^  (^l-l  iX  -5/  . 

— 60*  What  is  the  amount,  at  compound  interest,  of  $100,  from 
Apr.  1,  1860,  to  Jan.  1,  1863^  at  5^  per  annum,  interest  paya- 
ble semi-annually?        j  Ih     iy^x-h 

51.  The  interest  on  a  note  for  2  y.  10  d.,  at  7%,  is  $141.94|; 
what  is  the  face  of  the  note  ?   i  f]  ^  f} 

52.  In  what  time  will  $98,  on  interest  at  7%,  amount  to 
$123.48?  1.7-   &,7://:i 

53.  What^principal  will  amount  to  $185.50  in  3  y.  9  m.  20  d., 

at7%?  /H\^^- 

54.  What  would  be  due  May  1,  1865,  on  a  note  for  |1000, 
dated  March  26,  1860,  at  8%  interest,  on  which  $200  were  paid 

,  ,  «t  the  end  of  each  year  from  the  date  of  the  note^^:^  ^  ^  ,;  -^  Ij 
^''^'^55.  N.  T.  Allen  bought,  June  8,  1861, 10  bales  of  eottii  oioth, 
14  pieces  in  a  bale,  43  yds.  in  a  piece,  at  8  cts.  per  yd.,  for  wiich 
be  gave  his  note  on  interest  at  6%.  On  the  4th  of  Nov.,  1863,  he 
sold  1  bale  at  30  cts.  a  yd.,  and  with  the  proceeds  made  part  pay- 
ment of  his  note.  On  the  3d  of  May,  1864,  he  sold  1  bale  at  40 
cts.,  and  paid  on  his  note  the  amount  he  received.  On  the  17th 
of  Sept.,  1864,  he  sold  the  remainder  at  60  cts.,  and  settled  the 
note.     What  did  he  gain  by  his  speculation  ?  -  ^  ,.  :  /  % 

5Q.  George  Rivers  bought  $1000  worth  oi  government  stock 
at  par,  bearing  7^q  %  interest  in  U.  S.  currency.  At  the  end  of 
3  years,  he  converted  it  into  five-twenty  6%  bonds,  interest  pay- 
able in  gold.  Gold  being  at  a  premium  of  100%,  does  he. gain 
or  lose  by  the  exchange,  andv,rhat%?  Ans.  Gains  annually  4^^%. 
57.  I  buy  United  States  ten-forty  bonds  at  par,  interest  being 
5^/c  per  annum,  payable  semi-annually  in  gold,  and  gold  being  at 
a  premium  of  150%.  What  rate  per  cent,  in  currency^,  payable 
annually,  do  my  proceeds  equal  ?  Ans.  12|^%. 


MISCELLANEOUS  EXAMPLES.  321 

58.  A  pays  $1075  for  United  States  five-twenty  6%  bonds, 
at  a  premium  of  7^%,  the  interest  on  the  bonds  being  paid  semi- 
annually in  gold.  If  the  average  premium  on  gold  be  112^, 
does  he  make  more  or  less,  and  how  much,  than  B,  who  invested 
an  equal  sum  in  railroad  stocks,  at  14%  below  par,  which  paid  a 
semi-annual  dividend  of  4%  ?    Ans.  $13.60  more,  semi-annually. 

59.  The  city  tax  of  Lowell  being  f  %,  and  the  state  and  county 
tax  .15%  ;  for  what  sum  is  Samuel  Lowe  taxed,  who  pays  $56.22, 
including  $1.50  poll-tax?    (^  .'  .,  / 

60.  An  agent,  who  purchased  a  lot  of  wheat,  forwarded  hia 
bill  for  $568.87^.  If  this  included  his  commission  of  2J-%  on 
the  purchase,  what  sum  was  paid  for  the  wheat  ?    ;  ■" 

61.  What  is  the  difference  between  the  true  and  bank  discount 
of  $700,  due  in  90  days,  where  the  legal  rate  is  7%?  C  /  r  "/' 

62.  Write  a  note  for  60  days,  for  which  you  could  get  $300  at 
a  bank,  discount  being  6%.  3o  S  ^/t^ 

63.  How  much  would  you  receive  from  a  bank,  June  12, 1860, 
for  a  note  of  $820,  dated  April  12,  1860,  payable  6  months  after 
^ate  ? 

64.  A  bookbinder  holds  a  bill  against  a  publisher  for  work  to 
the  amount  of  $600,  payable  in  6  months  without  interest.  He 
offers  to  discount  5%  of  the  bill  for  present  payment.  If  the 
publisher  pays  $300  of  the  debt,  what  will  still  be  due  ? 

Ans.  $284.21+. 

65.  The  government  tax  on  all  bank  dividends  for  1864  was 
3%.  A  certain  bank,  having  declared  a  dividend  of  $15000, 
paid  to  the  government  $450,  and  subsequently  paid  the  full  sum 
of  $15000  to  the  stockholders.  To  this  the  government  objected. 
What  was  the  error  at  the  bank  ?  What  sum  should  have  been 
paid  to  the  government  iV^'—  r  !  '^^ 

QQ.  Blake  Brothers  &  Co.  purchased  to  order  oil  stocks  to  the 
amount  of  $5714.25,  including  their  commission  of  |%;  the 
stock,  the  par  value  of  which  was  $50  per  share,  was  purchased 
at  95%  ;  how  many  shares  were  purchased  ?  Ans.  120  shares. 
i  67.  Received  from  India  75  tierces  of  rice,  invoiced  at  220  lbs. 
each,  for  which  I  paid  3  cents  a  pound.  A  duty  of  2  cents  per 
21 


322  MISCELLANM)US  EXAMPLES. 

pound  was  paid  at  the  custom  house,  after  h^^/o  oi  the  weight  had 
been  deducted  for  tare.  If  the  rice  should  net  what  it  was  in- 
voiced at,  for  what  should  it  be  sold  to  gain  20%?y'V,^\   /•    I  k- 

68.  A,  B,  and  C  formed  a  partnership.     A  furnished  |  of  the 

capital,  B  ^,  and  C  the  remainder.     Their  gains  were  equal  to 

12^%  of  their  capital.     Of  these  C  took  25^  eagles  as  his  share.     ' 

Allowing  a  premium  fcMr  the  gold  of  24^^,  what  was  the  whole 

I,  iteipitaljjapd  )vh^.the  gain  of  A  and  B  in  bank  notes  ?^         /^^^,    > 

C9.  Bre(M  and  lioring  traded  in  liideS  for  one  ydar'.'  ^Brbck  put^ 
in  $2000  at  first ;  at  the  end  of  3  months  he  withdrew  $700, 
and  at  the  end  of  7  months  put  in  $1000.  Loring  put  in  $1200 
at  first,  and  $500  more  in  4  months.  At  the  end  of  6  months 
he  withdrew  $200.  The  gain  for  the  year  was  $2355.75,  of 
which  Loring  received  $1000  for  conducting  the  business.  What 
was  the  share  of  each  ?  ,>     ,,^,  -  ^/  />^<^</^ 

70.  What  should  be  the  date  of  a  note  given  in  payment  of 
the  balance  of  the  following  account  ? 

i)r.  Miles  Standish  ii^  ^  with  P.  Sinon.  Gr, 


1859.1  I  I      I 

May  14 1  To  balance  old  acc't,  $960  30! 
July    8 1    ''  Mdse., I   ol9|00i 


1859. 
Jan.  11 
Sept.  8 


By  Mdse., 

"  Cash, 


$800  GO 
475  60 


Am.  May  5,  1860. 

71.  Required  the  cost  of  bricks  to  build  the  walls  of  a  store- 
house 25  ft.  long,  20  ft.  wide,  and  30  ft.  high,  containing  2  win- 
dows 8  ft.  by  4  ft.  each,  and  1  door  7  ft.  by  6  ft,  the  walls  being 
2  ft.  thick,  and  bricks  $5.50  per  thousand,  measuring  8  in.  by 
4  in.  by  2  in.        (->?  f,  I    '  ' 

72.  Required  the  cost  of  boards,  at  $20  per  thousand  feet,  to 
make  a  box  7  it.  10  in.  long,  3  ft.  8  in.  wide,  and  2  it.  6  in.  high ; 
boards  to  be  1  in.  thick.  Ans.  $2.20§. 

73.  A  wine  merchant  used  the  following  receipt  for  port  wine: 
35  gallons  prepared  cider,  worth  $1.00  per  gallon  ;  5  gallons  red 
wine,  at  $2.00  per  gallon  ;  5  gallons  port  wine,  at  $5.00  per  gal- 
lon ;  3  gallons  spirits,  at  $1.00  a  gallon  ;  3  pounds  sugar,  at  16 
cents  per  pound  ;  2  ounces  tincture  of  kino,  at  6  cents  an  ounce ; 


MISCELLANEOUS||;XAMPLES.  323 

i 
and  Leunce  tartaric  acid,  at  13  cents.     Suppose  the  sugar,  kino, 
and  acid  do  not  add  to  the  bulk  of  the  mixture,  and  that  the  mer- 
chant  sells  it  at  $4.50  per  gallon,  what  per  cent,  does  he  gain  ?„  ,-j  )jiV 
74.  If  a  pipe  2^  inches  in  diameter,  will  fill  a  cistern  in  two   ''    ' 
hours,  in  what  time  wilra'prp^'S'iiffcheii  in  cl^ame'ter  fill  the  same?    0 
,—  75.  AVhat  is  the  length  of  the  edge  of  the  largest  cube  that  can 
be  sawed  from  a  globe  9  inches  in  diame^'?     ly  ^  j   'I   -- 

76.  The  ridge-pole  of  a  house  is  46  ft.  from  the  ground,  th* 
eaves  38  ft.,  the  rafters  on  each  side  of  the  roof  being  18  ft.  long; 
what  is  the  width  of  the  house? 

77.  Required  the  edge  of  a  cubical  box  that  will  contain  12a 
times  as  much  as  a  box  measuring  1  foot  each  way.    ' 

78.  The    pyramid  of  Cheops,  in   Egypt,  is    said  to  contaiiv 
82111000  cubic  feet  of  masonry,  and  to  ha'Te  been  480  feet  high. 
Allowing  7000000  cubic  feet,  which  are  required  to  perfect  its 
pyramidal  form  and  to  fill  its  cliambers,  whfft  is  the  length  of  ont*         ' 
side  of  its  base,  which  is  a  square  ?  Ans.  746.2  ft.-j-. 

79.  How  many  yards  of  cloth  |  yd.  wide  will  be  required  i& 
cover  the  sides  and  top  of  a  cubical  box  containing  6751.269 
cubic  inches  ? 

80.  What  will  be  the  cost  of  digging  a  ditch  outside  a  square 
garden  containing  12.75  rods,  the  ditch  to  be  7  ft.  wide  and  5  fit, 
deep,  at   1    cent  per  cubic  foot  ?    ;  ■ 

81.  How  much  would  the  earth  taken  out  of  the  ditch  raise  the       ^ 
surface  of  the  garden  ?  ,    [  -f- 

82.  How  many  gallons  in  a  cylindrical  jat,  the  radius  of  whose 
base  is  1  foot,  and  whose  altitude  is  4  ft.  ?   /  '-Ul      ,    JJ 

83.  "What  will  be  the  thickness  of  a  square  stick  of  timber 
which  contains  4.542098  tons  (50  cu.  ft.  =-z  1  ton),  the  stick  being 
100  ft.  long? 

84.  Supposing  a  cubic  foot  of  snow  to  weigh  31  lbs.,  what  will 
be  the  pressure  of  a  body  of  snow  9  inches  deep  upon  a  flat  roof 
100  fl.  by  25  fl.  ?     ^  n        I       I     / 

85.  Required  the  number  of  square  feet  in  the  surface  of  a 
ditch  surrounding  a  circular  garden  which  is  25  yards  across,  the 
ditch  being  24-  ft.  across.        L,  l     ':i  'k 


324  miscellan;^ug  examples. 

8G.  How  many  gallons  v/ill  the  above  ditch  contain,  it  being 
o  ft.  deep  ?    ^^-^  i  h'  '        ■.    -•  /' 

87.  The  circular  outlet  to  a  cistern  being  4  inches  in  diameter, 
■vvhat  must  be  the  width  of  a  rectangular  receiving-pipe,  whose 
depth  is  2  inches,  that  its  capacity  may  be  the  same  as  the  dis- 
charging-pipe  ?     . 

"■■^88.  What  will  a  pine  log  weigh  whose  length  is  18  ft.,  meas- 
\iring  3  ft.  across  the  larger  end,  and  2^  ft.  across  the  smaller, 
pine  being  .6  as  heavy  as  water,  which  weighs  62 J  lbs.  to  a  cubic 
foot  ?     (See  Art.  455.) 

^  89.  How  many  cubic  yards  in  a  cellar  whose  side  walls  meas« 
lire,  on  the  outside,  70  ft.,  and  whose  end  walls  measure  48  ft., 
the  cellar  being  10  ft.  deep,  and  the  walls  3  ft.  thick  7 //  ^  4, 

'90.  Suppose  there  is  a  globe  of  ice  in  the  region  of  the  Alps 
weighing  243474  lbs.;  there  being  930  oz.  to  a  solid  foot,  what  is 
its  diameter  ?  '  Ans.  20  ft. 

91.  Suppose  ^  of  the  above  globe  of  ice  to  melt  away  each 
yeaiv,>v.hat  will  be  the  length  of  its  diameter  each  succeeding 
year  ?**/  k  r 

92.  An  engineer  planted  a  battery  near  the  bank  of  a  river  to 
shell  a  fort  upon  the  opposite  side.  To  ascertain  the  distance  of 
the  fort,  he  noted  the  direction  of  the  fort  from  the  mortar ;  then, 
placing  himself  at  a  point  eight  rods  higher  up  the  river,  he  caused 
a  line  to  be  drawn  from  a  point  six  feet  distant  from  himself,  ia 
range  with  the  mortar,  to  be  extended  parallel  with  the  line  first 
noted  till  it  ranged  between  himself  and  the  fort.  This  line 
he  found  to  be  480  feet.  What  was  the  distance  of  the  fort  from 
the  mortar  ?    (Page  294,  Note.)  Ans.  2  miles. 

93.  Wishing  to  know  the  height  of  a  flagstaff  which  was  60 
feet  distant,  I  held  my  cane  perpendicularly  so  that  its  lower  end 
was  2  ft.  4  in.  in  a  horizontal  line  from  my  eye,  and  found  the 
range  of  the  top  of  the  staff  was  35  inches  from  the  bottom  of 
the  cane.  Required  the  height  of  the  flagstaff,  allowing  my 
eye  to  have  been  5  ft.  from  the  ground,  which  was  a  horizontal 
plane,  Ans.  8)  ft.  high. 


APPENDIX. 


SOME    OF   THE    PROPERTIES    OF   9. 


«511.  Any  number  may  he  separated  into  two  parts,  one  of 
which  is  divisihle  hy  9,  and  the  other  of  which  is  equal  to  the  sum 
of  its  digits. 

Illustration.     Let  5864  be  the  number  considered. 


5000  =  5  X  (999 


I  H-  60  =  6  X 
l-l-     4  = 


1)  —  5  X  999  +  5 

i)z=8X     99  —  8 

(9-j-l)=:6X       9  +  6 

4 


5864=:  (5X  999  +  8  X  99  +  6  X  9)  + (5  +  8  +  6^  4). 

.-.5864  is  separated  into  two  parts,  the  first  (5  X  999  +  8  X  99  +  6 
X  9)  being  divisible  by  9,  and  the  second  (5  +  8  +  6  +  4)  being  the 
sum  of  its  digits.     The  same  can  be  shown  of  any  number. 

The  following  principles  are  derived  directly  from  Art.  511 :  — 
31^.    Any  number  is  divisible  hy  9,  if  the  sum  of  its  digits  is 

divisible  by  9. 

•ilS*    If  any  number  is  divided  hy  9,  the  remainder  is  equal 

to  the  remainder  when  the  sum  of  its  digits  is  divided  hy  9. 

^i4:.     Pkoof  of   Multiplication  by  casting   out   the 
9's.     (See  Art.  50.) 
Proof. 
3  +  2  +  6  =  0+2 
4  +2=_6 

12.    1  +  2  r=  3,  remainder. 
1  +  3  +  6+2  =  12  =  0  +  3,  remainder^ 
These  remainders  being  equal,  the  work  is 
probably  correct. 

(325) 


Multiplication. 

326 
42 

652 
1304 


13692 


326  APPENDIX. 

Demonstration  of  Proof. 
326  -f-  9  leaves  an  excess  of  2.  We  separate  the  mul- 

42  -^  9  leaves  an  excess  of  6.  tiplicand  and  multiplier 

Q2g 094.   I    2      each  into  two  parts,  the 

4.2  :::z    36-4-6      ^^^^  part  being  divisible 

324X6  +  2X6  ^^  ^'  ^''^  ^^^  '^""''^ 

324  X364-2X36  P^^^  hemg  the  excess  of 

— ; 9's.     Multiplying-  these 

324  X  36  +  324  X  6  +  2  X  36  +  2  X  6  „„^,^,.^  ^^  =^p^^^. 

ted,  we  obtain  four  terms  for  a  product,  the  first  three  of  which  are 
divisible  by  9,  and  the  last  is  the  product  of  the  two  excesses.  The 
entire  product  divided  by  9  must,  therefore,  leave  the  same  remainder 
as  the  product  of  the  excesses  in  the  multipHcand  and  multiplie/ 
divided  by  9. 

515,     Proof   of   Division   by   casting   out   the   9's. 
(See  Art.  62.) 
Division.  •  Proof. 

75 )  3929  (52  7_)_5  =  12=:0+3 

375  5  +  2=         7^ 

"l79  21     2  + 1  =  3,  remainder. 

150  3929—29  =  3000, 3,  remainder. 

29  These  remainders  being  equal,  the  work  is  prob- 

ably correct. 

Demonstration  of  Proof. 

The  dividend,  minus  the  remainder,  equals  the  product  of  the  divi- 
sor and  integral  part  of  the  quotient ;  therefore,  it  v  e  divide  the  divi- 
dend, minus  the  remainder,  by  9,  the  remainder  thus  obtained  must 
be  the  same  as  that  which  results  from  dividing  the  product  of  the 
excess  of  9's  in  the  divisor  and  the  integral  part  of  the  quotient  by  9. 


CONTRACTIONS  IN  MULTIPLICATION  AND  DIVISION. 

Arithmetical  operations  may  sometimes  be  shortened  mate- 
rially by  the  use  of  contractions  in  Multiplication  and  Division. 
A  few  have  been  suggested  in  Articles  52,  53,  and  64.  Some 
additional  contractions  are  here  given,  which  pupils  are  cautioned 
against  using  until  they  are  so  familiar  with  the  common  methods 
as  to  make  no  mistakes. 


APPENDIX.  327 

516.     To  Multiply  by  9,  99,  999,  &c. 

9  being  one  less  than  10,  99  one  less  than  100,  and  999  one 
less  than  1000,  &c., 

To  multiply  by  any  number  whose  terms  are  all  9's :  Annex 
as  many  zeros  to  the  ^hultiplicand  as  there  are  9's  ifi  the  multiplier^ 
and  from  that  product  subtract  the  multiplicand;  thus,  27  X  99 
=  2700—27  =  2673. 


Examples. 


1.  36  X  99=? 

2.  264  X  999  =  ? 

3.  58  X  9999=? 


4.  36841  X  9999? 

5.  7  X  9999999  =  ? 

6.  245  X  999999  =  ? 


7.  241  X  998  =  ?      (241  X  998  =  241  X  1000  —  241  X  2.) 

Ans.  240518. 

8.  356  X  9995  =  ?  |      9.  54932  X  999997  =  ? 

517,  To  Multiply  by  a  Composite  Number,  i.  e.,  by  a 
Number  that  is  itself  the  Product  of  two  or  more 
Numbers. 

Separate  the  multiplier  into  convenient  factors,  multiply  the 
midtiplicand  hy  one  of  the  factors,  and  that  product  by  another 
factor,  and  so  on,  till  all  the  factors  are  employed ;  the  last  prod* 
uct  is  the  true  answer  ;  thus,  41  X  25  :=  41  X  5  X  5. 

Examples. 

1.  Multiply  368  by  72 ;  by  36. 

2.  Multiply  4079  by  81;  by  48. 

3.  Multiply  2145  by  108  ;  by  144. 

4.  Multiply  50411  by  55  ;  by  150. 

518.     To   Multiply   by   Aliquot   Parts   of   10,   100, 

1000,  &c. 
Multiply  hy  10,  100,  1000,  <^c.,  as  the  case  may  require,  and 
then  find  the  required  part ;  thus,  to  multiply  by  5,  multiply  by 
10,  and  divide  the  product  by  2 ;  to  multiply  by  25,  multiply  by 


828  APPENDIX. 

100,  and  divide  by  4  ;  by  125,  multiply  by  1000,  and  divide  by 
8;  by  33^,  multiply  by  100,  and  divide  by  3  ;  by  IGf,  multiply 
by  100,  and  divide  by  G ;  by  12^,  multiply  by  100,  and  divide 
by  8. 

Examples. 
1.  8743008  X  5  =  what? 


2.  8003478  X  25  =  what  ? 

3.  786342  X  12^  =  what  ? 


4.  875402  X  3^  =  what? 

5.  1090806  X  16|=  what  ? 

6.  543297  X  125  =  what  ? 


A 


^19*     To  Divide  by  a  Composite  Number. 
III.  Ex.     Divide  390  by  15. 

Operation.  To  divide  by  a  composite  number :  Sepa- 

3  )  390  rate  the  divisor  into  convenient  factors,  divide 

5  )  130  hy  one  factory  and  the  quotient  thus  obtained  by 

26,  Ans.      aii-other  factor,  and  so  on  till  all  the  factors  are 

employed.     The  last  quotient  is  the  answer  required. 


Examples. 


1.  Divide  243873  by  32. 

2.  Divide  39726  by  18. 


3.  8954^121  =  what? 

4.  49176 -^- 72  z=  what? 


«S^0.     To  find  the  True  Remainder. 

III.  Ex.     Divide  83248  by  84. 

Operation.  In  this  example  we  have  remain- 

3  )  S3242  Rem.  ders  after  the   several  divisions. 

4)27747     1.                          1  The  first  remainder  is  of  the  same 

7  )  6936     3.    3  X    3  =    9  denomination  as  the  first  dividend, 

990     6      6X12  —  72  °^  units.     The  second  remainder 

rr>                 •    -I      "TTZT  ^  of  the  Same  denomination  as  the 

1  rue  remainder,  82  i  v  •  i     j        o-       o  .u 

'  second  dividend,  or  3  s.     3  threes 

p:=.  9  units.  The  third  remainder  is  of  the  same  denomination  as  the 
third  dividend  or  12's.  6  twelves  :=  72  units.  The  entire  remainder 
equals  the  sum  of  these  several  remainders,  or  1 -[-9-f-72i=82. 
Hence, 

To  find  the  true  remainder :  Commence  with  the  remainder 
resulting  from  the  second  division,  and  multiply  each  partial  re- 
mainder  by  all  the  preceding  divisors  except  the  one  which  gave 
that  remainder,  and  add  the  sum  of  the  products  to  the  remain- 
h'r  resulting  from  the  first  division. 


(if'r 


APPENDIX.  329 


Examples. 


1.  86543-^-117  =  what? 

2.  234567  -^  324  =  what  ? 

3.  359762-^187  =  what? 


4.  32947-^-132  =  what? 

5.  927638 -f- 2800  =  what? 

6.  7362851  -^  693  =  what? 


•521.     To  Divide  by  Aliquot  Parts  op  10,  100,  1000,  &c. 

To  divide  by  5,  divide  by  10,  and  multiply  the  quotient  by  2 ; 
to  divide  by  25,  divide  by  100,  and  multiply  by  4;  by  125, 
divide  by  1000,  and  multiply  by  8  ;  by  33^,  divide  by  100,  and 
multiply  by  3 ;  by  16§,  divide  by  100,  and  multijjly  by  6 ;  by 
166§,  divide  by  1000,  and  multiply  by  6,  &c. 


Examples, 

1.  9876  -^  25  =  ? 

2.  34543-^-125=? 

3.  87096 -M6f  =  ? 


4.  432872  ^12i  =  ? 

5.  687904-^250=? 

6.  110748 -M662  =  ? 


MODES  OF  ESTIMATING  THE  TIME  BETWEEN  TWO  DATES. 

5^fi,  The  mode  adopted  in  this  book  for  estimating  the  time 
between  two  dates,  when  interest  is  computed  in  months  and 
days,  is  in  common  use  among  business  men,  and  is  the  one  most 
consistent  with  the  ordinary  method  of  computing  interest  at  30 
days  to  the  month. 

^23*  Another  mode  of  estimating  the  time  between  two 
dates,  is  to  find  the  number  of  entike  calendar  months,  and  count 
Jhe  remaining  days. 

«]»S4.  A  third  mode  of  finding  the  time  between  two  dates, 
consists  in  counting  the  exact  days  from  one  date  to  the  other. 

ILLUSTEATION. 

By  the  first  of  these  modes,  if  a  note  for  months,  which  matures* 
Feb.  10th,  1865,  were  discounted  at  a  bank  the  11th  of  December 
previous,  the  time  would  be  estimated  at  1  day  less  than  two  months  j 
viz.,  1  month  and  29  days. 

By  the  second  mode,  it  would  be  estimated  at  1  month  and  30  days, 
i.  e.y  2  months. 

♦  A  note  is  said  to  mature  when  it  becomes  due. 


330  APPENDIX. 

By  the  third  mode,  it  would'  be  estfuiated"  at  61  days,  or  2  months 
and  1  day. 

In  the  above,  and  in  all  similar  cases,  the  second  of  these  methods 
will  give  more  time  than  the  first,  and  the  third  will  give  more  time 
than  the  first  or  second. 

If,  however,  the  above  note  matured  March  10,  and  were  discounted 
Jan.  11,  whilst  the  time  by  the  first  mode  would  be  1  day  less  than  2 
months,  —  viz.,  1  month,  29  days,  —  by  the  second  mode  it  would  be 
1  month,  27  days,  and  by  the  third  mode,  58  days,  cr  1  month,  28 
days.  In  this  case,  the  third  mode  gives  less  time  than  the  first,  and 
the  second,  less  time  than  the  third  or  first.  These  differences  arise 
from  the  difference  in  the  length  of  the  calendar  months  ;  and,  since  a 
majority  of  the  months  have  31  days,  the  second  and  third  modes  will, 
on  the  whole,  give  more  time  than  the  first,  and  the  third  more  time 
than  the  first  or  second. 

To  avoid  these  irregularities,  it  is  customary  to  make  notes,  running 
for  short  times,  payable  in  30,  60,  or  90  days,  instead  of  1,  2,  or  3 
months. 

Note  1.  —  By  custom,  at  the  banks,  a  note  which  is  given  for  months 
matures  on  the  day  corresponding  with  its  date :  if  the  month  in  which 
it  matures  has  no  corresponding  day,  it  matures  on  the  last  day  of  the 
month.  Thus,  four  notes  dated  severally  the  28th,  29th,  30th,  and  31st 
of  Dec,  1864,  and  given  for  two  months,  all  mature  Feb.  28,  with  grace, 
March  3 ;  while  one  dated  Feb.  28  would  mature  April  28  and  May  1. 

Note  2.  —  A  note  falling  due  on  the  Sabbath,  or  on  a  legal  holiday, 
must  be  paid  on  the  business  day  next  preceding.  Thus,  when  a  holiday 
occurs  on  Monday,  notes  maturing  that  day  must  be  paid  on  the  pre- 
vious Saturday. 

To  Compute  Interest  for  Exact  Days. 

S^5»  The  interest  for  days  is  calculated,  in  some  of  the 
tlnited  States,  in  Great  Britain,  and  by  the  United  States  gov- 
ernment, at  365  days  to  the  year;  1  day's  interest  being  consid- 
ered al^  of  a  year's  interest.  By  the  ordinary  method  it  is 
calculated  at  360  days  to  the  year;  this  gives  a  year's  interest 
/or  gea  of  a  year,  which  is  ^f  5,  or  j^^  too  much.     Hence, 

To  obtain  the  true  interest  for  days  :  Subtract  from  the  interest 
found  by  the  ordinary  method,  -^  of  itself , 


APPENDIX. 


831 


A  TABLE 
Showing  the  Number  of  Days 


FROM  ANY 

.  TO  THE  SAME  DAY  OF 

DAY  OF 

Jan. 

Feb. 

Mar. 

Apr. 

May. 

June. 

July. 

Aug. 

Sept. 

Oct. 

Nov. 

Deo. 

January, 

365 

31 

59 

90 

120 

151 

181 

212 

243 

273 

304 

334 

February, 

334 

365 

28 

59 

89 

120 

150 

181 

212 

242 

273 

303 

March, 

306 

337 

365 

31 

61 

92 

122 

153 

184 

214 

245 

275 

April, 

275 

306 

334 

365 

30 

61 

91 

122 

153 

183 

214 

244 

May, 

245 

276 

304 

335 

365 

31 

61 

92 

123 

153 

184 

214 

June, 

214 

245 

273 

304 

334 

365 

30 

61 

92 

122 

153 

183 

July, 

184 

215 

243 

274 

304 

335 

365 

31 

62 

92 

123 

153 

August, 

153 

184 

212 

243 

273 

304 

334 

365 

31 

61 

92 

122 

September, 

122 

153 

181 

212 

242 

273 

303 

334 

365 

30 

61 

91 

October, 

92 

123 

151 

182 

212 

243 

273 

304 

335 

365 

31 

61 

November, 

61 

92 

120 

151 

181 

212 

242 

273 

304 

334 

365 

30 

December, 

31 

62 

90 

121 

151 

182 

212 

243 

274 

304 

335 

365 

Note.  —  In  leap  years,  if  the  last  day  of  February  is  included  in  the 
time,  a  day  must  be  added  to  the  number  obtained  from  the  table. 


MEASUREMENT    OF   LUMBER. 


^^ 


»30.  The  contents  of  boards,  and  of  hewn  and  round  timber, 
whether  they  are  of  uniform  dimensions  throughout  or  taper  reg- 
ularly, may  be  found  by  rules  explained  in  Mensuration.  The 
following  additional  directions  for  finding  their  contents  may  be 
serviceable :  — 

Board  Measure. 

^ST,  If  a  board  be  1  inch  or  less  in  thickness,  its  contents 
are  found  hy  multiplying  its  length  hy  its  mean  breadth.  The 
mean  breadth  of  a  board  tapering  regularly  is  half  the  sum  of  the 
breadth  of  its  two  ends.  If  it  is  irregular  in  shape,  the  average  of 
a  number  of  measurements  at  equal  distances  must  be  used  as  the 
mean  breadth.     If  the  board  is  more  than  1    inch   thick,  the 


832  APPENDIX. 

square  contents  in  feet  must  he  multiplied  hy  the  thickness  in 
inches:  Thus,  a  board  whose  length  is  7  feet,  mean  breadth  2 
feet,  and  thickness  ^  inch,  contains  14  feet,  board  measure; 
but  a  board  whose  length  and  breadth  are  the  same  as  given 
above,  and  whose  thickness  is  2  inches,  contains  28  feet,  board 
measure. 

Examples. 

1.  Required  the  contents  of  a  board  10  ft.  long  and  1  in.  thick, 
which  is  2  ft.  wide  at  one  end,  and  1  ft.  6  in.  wide  at  the  other. 

2.  Required  the  contents  of  a  board  14  ft.  long,  J  in.  thick,  and 
measuring  at  the  ends  2  ft.  3  in.  and  1  ft.  9  in.  respectively. 

3.  Required  the  contents  of  a  board  16  ft.  long  and  1^  in.  thick, 
Tvhose  mean  breadth  is  2  ft.  5  in. 

MEASUREMENT   OF   TIMBER. 

528.  The  contents  of  timber  are  often  estimated  in  board 
measure,  and  are  found  bi/  multiplying  the  product  of  the  length 
and  mean  breadth  expressed  in  feet,  hy  the  mean  depth  expressed 
in  inches.  If  estimated  in  cubic  measure,  the  product  of  the 
length,  mean  hreadth,  and  mean  depth,  expressed  in  the  same 
dimensions,  must  he  found. 

Examples. 

1.  Required  the  contents  of  a  piece  of  joist  20  ft.  long,  whose 
mean  breadth  is  4  inches,  and  mean  depth  3  inches,  board  meas- 
ure.     '^  i 

^^2.  Give  the  contents  in  cubic  measure.  /  ;   '  =^  /  '  ^ 

^/3.  How  many  feet,  board  measure,  in  a  stick  of  timber  15  ft. 
long,  the  breadth  at  the  ends  being  6  in.  and  4  in.,  and  the  depth 
at  the  ends  4  in.  and  2^  in.  respectively  ? 

5S^,  The  contents  of  round  timber  may  be  obtained  hy  as- 
certaining  its  mean  diameter,  andfro?n  that,  as  a  hasis,  estimating 
its  contents  as  if  it  were  a  cylinder,  or  hy  squaring  I  of  the  mean 
girt,  and  multiplying  the  square  hy  twice  the  length. 

1.  How  many  cubic  feet  in  a  stick  of  round  timber,  whose 
mean  gh-t  is  8  ft.,  and  whose  length  is  12  ft. .'' 


APPENDIX.  r»33 


MISCELLANEOUS. 

^SOa  Shingling,  and  other  plain  work,  as  flooring  and  par- 
titioning, are  generally  estimated  by  the  square  of  100  feet.  1000 
shingles  are  allowed  for  a  square. 

«$<ll«    Painting  is  measured  by  the  square  yard. 

•53S,  Plastering  is  measured  by  the  square  foot,  square 
yard,  or  square  of  100  ft. 

«l33o  Glazing  is  measured  by  the  square  foot,  including 
the  sash. 

5S^>  Paving  is  measured  by  the  square  foot,  or  square 
yard. 

^S5*  Bricklaying  is  generally  estimated  by  the  thousand 
bricks ;  sometimes  it  is  estimated  by  the  square  yard,  square  rod, 
or  square  (of  100  ft.),  allowing  1^  bricks,  or  12  in.  in  thickness. 

A  great  variety  of  methods  for  measuring  prevail.  Some 
workmen  make  no  allowance  for  doors  and  windows,  others  make 
allowance  of  half  the  space  occupied  by  doors  and  windows,  and 
others  still  estimate  the  exact  amount  of  material  and  labor  em- 
ployed. Measurements  are  taken  on  the  outside  of  walls,  no 
allowance  being  made  for  corners.  In  estimating  the  number  of 
bricks  used,  an  allowance  of  one  tenth  of  the  solid  contents  is 
made  for  mortar. 

GAUGING. 

•>36*  Gauging  is  the  process  of  finding  the  capacity  of 
casks,  or  other  vessels,  in  gallons  or  bushels. 

cSvlT.    To  find  the  capacity  of  a  cask  or  barrel : 

Add  the  squares  of  the  head  diameter,  of  the  bung  diameter,  and 
of  twice  the  middle  diameter.  Multiply  the  sum  thus  obtained  by 
.0005667 ^me5  the  length,and theresult  willbethe contents  ingaUons. 

Note.  —  The  middle  diameter  is  the  diameter  of  the  section  midway 
between  the  bung  and  the  head.  The  dimensions  used  in  the  above  rule 
should  be  expressed  in  inches. 

1.  Find  the  capacity,  in  gallons,  of  a  cask  whose  length  is 
40  inches,  the  head  diameter  25  inches,  the  bung  diameter  82 
inches,  and  the  middle  diameter  29  inches.    Ans.  113.634  gal.-f- 


334  APPENDIX. 

New    Hampsiiike    Rule    fou    Annual    Interest    with 
Partial  Payments. 

t5S8.  AVhen  notes  are  given  upon  "annual  interest"  (Art. 
277),  and  partial  payments  are  made  during  the  year,  instead 
of  following  the  United  States  rule  (Art.  274),  which  makes  a 
new  principal  at  the  time  of  each  payment,  — 

1.  Compute  annual  interest  upon  the  princijoal  to  the  end  of  the  first 
year  in  which  any  payments  are  made ;  also  compute  interest  upon  the 
payment  or  j^o^ynients  from  the  time  they  are  made  to  the  end  of  the 
year. 

2.  Apply  the  amount  of  such  payment  or  payments,  frst  to  cancel 
any  interest  that  7nay  have  accrued  upon  the  yearly  interests,  then  to 
cancel  the  yearly  interests  themselves,  and  then  toioards  the  payment 
of  the  principal. 

3.  Proceed  in  the  same  icay  toith  succeeding  jyayments. 

4.  If,  at  the  time  of  any  payment,  no  interest  is  clue  except  what  is 
accruing  during  the  year,  and  the  payment  or  payments  are  less  than 
the  interest  due  at  the  end  of  the  year,  deduct  such  payment  or  pay- 
ments at  the  end  of  the  year,  loithout  interest  added. 

Note.    No  interest  should  be  computed  beyond  the  time  of  settlement. 

^00.     Examples. 

1.  A  note  for  $2500,  dated  Oct.  1,  1860,  was  given  on  demand, 
with  interest  payable  annually  at  6  %. 

r  June       1,  1862,   $500.00. 
Endorsements,  \  March  17,  1863,       87.94. 
(jDec.       1,  1865,    1000.00. 
What  was  due  April  1,  1867  ? 

Operation. 
Principal,  $2500.00  Oct.  1,  1860. 

Yearlv  5  i"te^est,  150.00  to  Oct.  1,  1861. 

'     ^  )  interest,  150.00  to  Oct.  1,  1862. 

Sunple  interest  on  {  ^  ^  5  from  Oct.  1,  1861, 


$150.00,  5      ^l!^  \      to  Oct.  1,  1862. 

Amount,  2809.00  Oct.  1,  1862. 

Payment,  500.00  June  1,  1862. 

Interest,  10.00  to  Oct.  1,  1862. 

Amount  of  payment,  510.00 

Principal,  2299.00  Oct.  1,  1862. 


APPENDIX. 

335 

PrjNCirAL  bro't  forw'd 

$2299.00 

Interest, 

137.94 

to  Oct.  1,  1863. 

Payment, 

87.94 

March  17,  1863. 

Balance  yearly  int., 

50.00 

due  Oct.  1,  1863. 

Simple  interest, 

9.00 

to  Oct.  1,  1866. 

C  interest. 

137.94 

to  Oct.  1,  1864. 

Yearly  j  interest, 

137.94 

to  Oct.  1,  1865. 

(  interest. 

137.94 

to  Oct.  1,  1866. 

$137.94  at  simple  int., 

24.83 

497.65 

for  2  yrs.  -\-  1  yr.  ==  3  yrs. 

Amomit, 

2796.65 

Payment, 

1000.00 

Dec.  1,  1865. 

Interest, 

50.00 

to  Oct.  1,  1866. 

Amount  of  payment,     ~ 

1050.00 

Principal, 

1746.65 

Oct.  1,  1866. 

Interest, 

52.40 

to  April  1,  1867. 

Balance  due, 

1799.05 

((             « 

Afis.  $1799.05. 

2.  A  note  for  $4200,  given  May  27,  1862,  was  payable  on 
demand,  with  interest  at  6  %  annually;  what  was  due  on  the 
note  May  27,  1867,  the  following  payments  having  been  made: 

Pavments  $  ^"--    ^'  ^^^^'      ^^^•'^2. 

raymenis,  j  j^^^  ^7,  1867,     3000.00.     Ans.  $2500.48. 

3.  A  note  given  for  $5000,  on  demand,  at  6  %  annual  interest, 

was  dated  January  1,  1860,  and  endorsed  as  follows: 

July  1,  1860,    $50.  July  1,  1865,  $500. 

"     1,  1861,    500.  "      1,  1866,  2500. 

What  remained  due  after  the  last  payment  was  made  ?     Ans.  $3703.62. 

040.     To  Compute  Interest  at  7y^^  %. 

Multiply  the  principal  hy  twice  the  numher  of  days,  and  point 
off  four  places  if  the  principal  contains  only  dollars^  six  places 
if  it  contains  cents. 

Examples. 

-   1.  What  is  the  simple  interest  of  $100  for  12  days,  at  7y=V  %  ? 

Ans.  $.24. 
.-^2.  What  is  the  simple  interest  of  $300  for  9  days,  at  7tV  %  ? 
!  Ans.  $.54. 

X 


336  THE  METKIC  SYSTEM. 


THE   METRIC   SYSTEM   OF   WEIGHTS  AND 
MEASURES. 

Note.  — The  Metric  System  of  weights  and  measures  was  first  adopted 
in  France  in  1795.  A  length  supposed  to  be  one  ten  millionth  of  a 
quadrant,  or  one  forty  millionth  of  a  circumference  of  the  earth  meas- 
ured over  the  poles,  was  taken  as  a  provisional  measure  for  the  base  of 
the  system ;  this  length  was  called  a  Meter. 

In  order  to  ascertain  more  accurately  the  length  of  a  quadrant,  new 
measurements  of  the  earth  were  subsequently  instituted  under  the  direc- 
tion of  eminent  mathematicians,  who  measured  the  arc  of  a  meridian 
between  the  parallels  of  Dunkirk  and  Barcelona.  From  their  measure- 
ments, the  length  of  the  meter  now  in  use  was  determined.  This  length 
was  adopted  as  the  base  of  the  system,  in  1799.  The  use  of  the  metric 
system  was  not,  however,  legally  enforced,  to  the  exclusion  of  any  other 
system,  until  January  1,  1840. 

In  Spain,  Portugal,  and  Belgium,  this  system  is  also  used  exclusively, 
while  in  many  other  countries  it  is  adopted  wholly  or  in  part.  Among 
tliese  are  Holland,  Italy,  Greece,  Austria,  Switzerland,  and  Poland,  in 
Europe,  and  Mexico,  Chili,  Venezuela,  Brazil,  Ecuador,  Guatemala, 
San  Salvador,  and  the  Argentine  Republic,  on  this  continent. 

Movements  are  also  being  made  to  adopt  it  in  England,  Germany, 
Sweden,  and  Norway.  Its  use  in  the  United  States  was  legalized  by  an 
act  of  Congress,  passed  in  July,  1866. 

Notwithstanding  so  much  has  been  done  to  make  the  meter  exactly 
one  ten  millionth  of  a  quadrant,  it  is  now  thought  to  be  too  short  by  a 
small  fraction,  which  is,  however,  less  than  one  eight  thousandth  of  an 
inch.  The  length  of  the  meter  is  nearly  39.37079  English  inches,  or 
39.3685  United  States  inches ;  but  for  ordinary  purposes,  may  be  con- 
sidered 39.37  inches. 

454:1  •  The  Metric  System  is  so  called  from  the  Meter,  which 
is  the  base  of  all  the  weights  and  measures  which  it  employs.     . 

The  Meter  is  the  primary  unit  of  length,  and  equals  about 
39.37  inches,  or  nearly  3  ft.  3|  in. 

Upon  the  Meter  are  based  the  following  primary  units :  the 
Square  Meter,  the  unit  of  measure  for  small  surfaces  ;  the  Are 
the  unit  of  land  measure  ;  the  Cubic  Meter,  or  Stere,  the  uni 
of  volume  ;  the  Liter,  the  unit  of  capacity  ;  and  the  Gram,  ih& 
unit  of  weight. 

From  these  primary  units  the  higher  and  lower  orders  of 
units  are  derived  decimally. 


i 


MEASURES  OF  LENGTH.  337 

54:^0  The  names  of  the  higher  orders  of  units  are  formed  by 
prefixing  to  the  name  of  tlie  primary  unit  the  following,  from 
the  Greek  numerals  :  — 

Deka  (10),  Hecto  (100),  Kilo  (1000),  Myria  (10000). 

The  names  of  the  lower  orders  of  units  are  formed  by  pre* 
fixing  to  the  name  of  the  primary  unit  the  following,  from  the 
Latin  numerals :  — 

Deci  (lOth),  Centi  (100th),  MilU  (1000th). 

Consequently,  the  word  dekameter  signifies  ten  meters  ;  deka- 
litevy  ten  liters  ;  hectometer,  one  hundred  meters ;  hectogram^ 
one  hundred  grams ;  kilometer,  one  thousand  meters ;  myria- 
meter,  ten  thousand  meters,  etc. 

So,  also,  the  word  decimeter  signifies  the  tenth  part  of  a 
meter  ;  centigram,  the  hundredth  part  of  a  gram  ;  milliliter^  the 
thousandth  part  of  a  liter,  etc. 

MEASURES  OF  LENGTH. 

Note.  —  In  this  table,  and  in  those  which  follow,  the  name  of  the  pri- 
mary unit  is  designated  by  capitals,  and  the  names  of  other  important 
units  by  italics. 

543.   Table. 

10  mil'limeters  Q"^'"^)  =.  1  centimeter,  marked  ("'"). 

10  cen'timeters  =  1  decimeter,  "  ^decim\ 

10  dec'imeters  =  1  meter,  "  (™). 

10  METERS  =1  dekameter,  "  ^dekam\ 

10  dek'ameters  =:  1  hectometer,  "  C"^)» 

10  hec'tometers  =  1  kilometer,  "  C^)' 

10  kil'ometers  =  1  myriameter,  "  /myriam\ 

Exercises. 

1.  How  many  meters  equal  1  dekameter ?/■' 1  hectometer? 
1  kilometer  h  q  1  myriameter  ? 

2.  How  many  dekameters  equal  1  hectometer?    1  kilometer? 

3.  1  meter  equals  how  many  decimeters?  how  many  centi- 
meters ?  how  many  millimeters  ? 

22 


338 


THE  METRIC  SYSTEM. 


K    IN' 

o 


—  o 


04:4:.  The  outer  diagram  in  the 
margin  represents  a  measure  4  inches 
in  length  ;  the  inner  diagram  represents 
a  measure  1  decimeter  or  10  centimeters 
in  length. 

These  diagrams  will  enable  the  pupil 
to  compare  the  units  of  length  of  the 
metric  system  with  those  in  common  use. 

Note  1.  — The  new  five- cent  piece  (of  1866) 
is  2  centuneters  hi  diameter.      # 

Note  2.  —  25  millimeters,  or  2-^  centimeters, 
nearly  equal  1  inch. 

Note  3.  —  5  meters  are  nearly  equal  to  1  rod. 

Note  4. —  1  kilometer  is  a  little  less  than  |^ 
of  a  mile. 

Note  5.  —  1  myriaraeter  is  a  little  more  than 
6^  miles. 

Although  the  meter  is  generally  con- 
sidered the  unit  of  length,  yet  in  esti- 
mating great  distances,  as  the  length  of 
a  road,  of  a  river,  the  distance  between 
two  cities,  etc.,  the  kilometer  is  regarded 
as  the  unit ;  thus,  the  length  of  the  Ohio 
River  is  1528  kilometers,  the  distance 
from  Troy  to  New  York  is  267  kilo- 
meters. 


54:5.   The  manner  of  writing  the  different  orders  of  units 
of  length  is  illustrated  by  the  following 


Table. 


1 

i 

i 

s 

P. 

1 

i 
1 

CO 

o 

p 

0) 

a 

B 

6 

1 

1 

1 

1 

1. 

1 

1 

1 

MEASURES  OF  LENGTH.  339 

In  writing  numbers  by  the  metric  system,  the  decimal  point 
is  usually  placed  at  the  right  of  the  figure  denoting  the  primary 
unit ;  thus,  the  number  5  meters,  9  decimeters,  is  written  5.9™. 

If  in  writing  a  number  any  intermediate  orders  of  units  are 
wanting,  their  places  should  be  supplied  by  zeros  ;  thus,  1  deka- 
meter,  2  millimeters,  is  written  10.002™. 

Exercises. 
Write  the  following  in  figures  :  — 
-1.  Three  (J^kameters,  four  meters.  Ans,  34™. 

^2.  Seven  hectometers,  three  meters.  Ans,  703™. 

^3.  Three  hectometers,  one  dekameter,  five  meters.  i  /  ^\, 

-^4.  Three  kilometers,  two  hectometers,  seven  meters.      3  ^ 
-  5.  Nine  myriameters,  five  hectometers.  /  ^  v  ^ 

--6.  Two  meters,  four  decimeters.  Ans.  2.4™. 

~  7.  Two  meters,  two  centimeters,  four  millimeters. 
""8.  Five  dekameters,  two  decimeters,  eight  centimeters. 

54LO,  When  numbers  are  expressed  by  figures,  the  part  of  the 
expression  at  the  left  of  the  decimal  point  is  usually  read  in  the 
denomination  of  the  primary  unit ;  the  part  at  the  right  of  the 
decimal  point  may  be  read  either  as  a  decimal  part  of  the  unit, 
or  in  the  denomination  indicated  by  the  place  of  the  last  figure. 
Thus,  in  reading  the  expression  34.62™,  we  may  say  either  34 
and  62  hundredths  meters,  or  34  meters  and  62  centimeters. 

Exercises. 
Read  the  following,  giving  the  name  of  each  order  of  units  :  — 
1.  23™.       2.  25.1™.       3.  321.05™.       4.  7137.008™. 

5.  Read  the  above  in  the  denomination  of  the  units  as  indi- 
cated by  the  abbreviation'™. 

6.  Read  the  same,  giving  each  decimal  part  the  denomination 
indicated  by  the  place  of  the  last  figure. 

Since  the  metric  system  is  a  decimal  system,  a  number 
expressed  in  units  of  one  order  may  be  reduced  to  units  of 
another  order  by  multiplying  or  dividing  by  ten,  or  some  power 


340  THE   METRIC   SYSTEM. 

of  tea.  If  the  number  is  written  in  figures,  it  is  only  neces- 
sary to  remove  the  decimal  point  to  the  right  of  the  figure 
indicating  the  required  order,  and  give  the  expression  its  proper 
abbreviation  ;  thus,  59.36'"  may  be  reduced  to  5.936^*=^^",  .59GG''''", 
.05936'^'",  .005936"^^"*'",  593. 6*^^""^,  593G."",  59360.""". 

547.     Examples. 

1.  Express  5.24  meters  as  centimeters.  Ans.  524*''". 

2.  Express  37.2  meters  as  kilometers.  Ans.  .0372^"*. 

3.  Express  12  hectometers  as  meters.  Ans.  1200™. 

4.  Express  25  millimeters  as  meters.  Ans,  .025"^. 

5.  In  518  meters  how  many  decimeters?  how  many  centi- 
meters ?  how  many  millimeters  ? 

6.  In  3687  metres  how  many  dekameters?  how  many  hec- 
tometers ?  how  many  kilometers  ? 

7.  Express  in  meters  the  following,  and  add  them:  4075 
centimeters,  27  dekameters,  .075  kilometers.         Ans.  385.75'". 

8.  Express  in  kilometers  the  following,  and  add  them  :  2400 
meters,  500  dekameters,  .79  myriameters.  Ans.  15.3^™. 

9.  If  7.08  kilometers  are  taken  from  42  kilometers,  how 
many  meters  will  remain  ?  Ans.  34,920". 

10.  The  distance  round  a  certain  park  is  2.58  kilometers ; 
how  many  meters  will  a  man  go  who  rides  around  it  six  times? 

Ans.  15,480"". 

11.  A  schoolboy  walked  one  third  around  the  above  park  in 
1 2  minutes  ;  how  many  meters  did  he  walk  in  1  minute  ? 

Ans.  71.66"^+. 

12.  The  latitude  of  Chicago  is  42°  N. ;  how  many  kilometers 
is  it  from  Chicago  to  the  equator?  Ans.  46 6 6. 6 e'""'-}-. 

V.  MEASURES   OF  SURFACE. 

5J4:8.  In  the  measurement  of  small  surfaces  the  Square 
Meter  is  the  primary  unit. 

Each  side  of  a  square  meter  is  10  decimeters  in  length,  and 
hence  a  square  meter  contains  10  X  10,  =  100,  square  deci- 
meters.    Each  side  of  a  square  decimeter  is  10  centimeters  i'^^ 


MEASURES  OF  SURFACE.  341 

length,  and  lience  a  square  decimeter  contains  10  X  10,=  100, 
square  centimeters,  etc.  Thus  it  may  be  seen  that  while  meas- 
iTi'es  of  length  increase  and  decrease  by  a  scale  of  tens,  measures 
of  surface  increase  and  decrease  by  a  scale  of  hundreds. 

Since  the  values  of  imits  of  surface  increase  and  decrease  by 
a  scale  of  hundreds,  it  is  necessary,  in  writing  numbers  denoting 
surfaces,  to  allow  tw  o  places  for  sq.  decimeters,  two  for  sq.  centir 
meters,  and  two  for  sq.  millimeters  :  thus, 

4  sq.  meters,  8  sq.  decimeters,  are  written  4.08^'^™, 
In  what  places  at  the  right  of  the  decimal  point  are  sq.  deci- 
meters written  ?  sq.  centimeters  ?  sq.  millimeters  ? 

Examples. 

1.  Express  the  following  numbers  in  sq.  meters  and  add  thpm  : 
5  sq.  decimeters,  87  sq.  meters,  26  sq.  centimeters,  5.9  sq. 
meters.  A71S.  92.9526^*^'". 

2.  How  many  sq.  meters  are  there  in  a  rectangular  court 
5  meters  long  and  22  meters  wide?  Ans.  110*'^'". 

3.  What  is  the  cost  of  polishing  the  surface  of  a  rectangular 
piece  of  marble  2  meters  8  decimeters  long,  and  1  meter  2 
decimeters  wide,  at  $2.50  per  sq.  meter?  Ans.  $8.40. 

049*  We  have  seen  that  the  meter  and  its  subdivisions  are 
used  to  measure  small  surfaces  ;  but  to  measure  surfaces  of  great 
extent,  as  a  field,  a  township,  etc.,  the  Are  is  the  primary  unit. 

The  Are  is  a  square  whose  side  is  10  meters  and  whose 
surface  contains  100  square  meters. 

In  land  measure,  centares,  ares,  and  hectares  only  are  used. 

Note.  — The  are  equals  119.6  square  yards,  nearly  4  square  rods,  or 
about  -^Q^  of  an  acre.     The  hectare  equals  about  2-^  acres. 

Table, 

100  cen'tares  (''')  =  1  are,      marked  ("). 
100  ARES  =   1  hectare        "        l^"). 

550,  The  following  table  shows  the  method  of  writing 
numbers  in  land  measure,  also  the  relation  of  the  units  to  the 
square  meter  and  its  subdivisions. 


342  THE  METRIC  SYSTEM. 

Table. 


cj  (u  a  2  a  s 

U  W^.  .^  'B  i 

O  I            "S^  ^  ^  g 

03  w         a»  o  . 

W  ^      o  S  ^  ^ 


1     0  1.01010101 

Examples. 

1.  Express  tlie  following  in  ares  and  add  them  :  1.3  hectares, 
155.5  ares,  43  hectares,  26  centares.  Ans.  4585.76^'". 

2.  Mr.  Jenks  owned  25  hectares,  32  ares,  16  centares  of 
land,  and  afterwards  bought  36  hectares,  5  ares,  8  centares  ; 
how  many  ares  did  he  then  have?  Ans.  6137. 24*^ 

3.  A  had  6  hectares,  7  ares,  9  centares  of  land,  and  sold  -^^ 
of  it  at  $54  an  are  ;  how  many  dollars  did  he  receive  for  what 
he  sold?  Ans,  $5960.52. 

r  MEASURES  OF  VOLUME. 

S51,  In  the  measurement  of  solids,  the  Cubic  Meter  is 
the  primary  unit. 

Each  edge  of  a  cubic  meter  is  10  decimeters  in  length,  and 
hence  a  cubic  meter  contains  10  X  10  X  10,  =  1000,  cubic 
decimeters.  Each  edge  of  a  cubic  decimeter  is  10  centimeters 
in  length,  and  hence  a  cubic  decimeter  contains  10  X  10  X  10, 
=  1000,  cubic  centimeters,  etc. 

Thus  it  may  be  seen  that  while  measures  of  length  increase 
and  decrease  by  a  scale  of  tens,  measures  of  volume  increase 
and  decrease  by  a  scale  of  thousands.  Hence,  in  writing  num- 
bers denoting  volume,  three  places  must  be  allowed  for  cu.  deci- 
meters, three  for  cu.  centimeters,  and  three  for  cu.  millimeters. 

In  what  places  at  the  right  of  the  decimal  point  are  cu.  deci- 
meters written?   cu.  centimeters?    cu.  millimeters? 

Examples. 
1.  Express  the  following  numbers  in  cu.  meters  and   add 
them:    58.5  cubic  meters,  1.7  decimeters.        ^ns.  58.5017*="". 


MEASURES  OF  VOLUME.  343 

"^    2.  How  many  cu.  meters  in  a  cube  whose  edge  is  2.7  meters? 

Ans.  19.683«""\ 
~  3.  How  many  cubic  meters  of  air  will  a  room  contain  whose 
length  is  5.2  meters,  whose  breadth  is  4  meters,  and  whose 
height  is  35  decimeters?  Ans.  72.8"^"  «". 

552»    For  measuring  firewood,  stone,  etc.,  the  Stere  is  the 
primary  unit.     Dekasteres  and  decisteres  are  also  used. 

The  stere  is  a  cubic  meter,  or  1.308  cubic  yards,  and  is  a 
little  more  than  ^  of  a  cord. 

Table. 
10  dec'isteres  C^')  =  1  stere,      marked  (*). 
10  STERES  >i/M,    =  1  dekastere       "       (^^^^^), 

The  method  of  writing  numbers  in  wood  measure  is  the  same 
as  that  of  writing  numbers  in  measures  of  length.    (Art.  545.) 
Examples. 

1.  How  many  steres  will  a  pile  of  wood  contain  that  is  1 
meter  long,  1  meter  wide,  and  1  meter  high  ?  2  meters  high  ? 

2.  What  part  of  a  stere  will  a  pile  of  wood  contain  that  is  1 
meter  long,  1  meter  wide,  and  1  decimeter  high  ? 

3.  How  many  steres  in  a  pile  of  stone  that  is  1  meter  wide, 
8.24  meters  long,  and  4  decimeters  high  ?  Ans.  3.296^ 

4.  What  must  be  the  height  of  a  pile  of  wood  2.5  meters 
long  and  1  meter  wide,  to  contain  a  stere  ?  Ans.  4*^^""'. 

MEASURES  OF  CAPACITY. 
S5S,    In  measuring  liquids,  as  milk,  and  dry  articles,  as 
beans,  barley,  and  salt,  the  Liter  is  the  primary  unit. 

The  Liter  is  1  cubic  decimeter,  and  contains  .908  of  a  quart 
dry  measure,  or  1.0567  quarts  liquid  measure. 

Table.  ^ 

10  mil'liliters  ('"^)    ==  1   centiliter,    marked  C'^).  // 

10  cen'tiliters '  =  1  deciliter,  "        (^«=").  ^^ 

10  dec'iliters  =  1  liter,  "        O-  ^ 

10  liters  =:  1  dekaliter,        "        (^^^*0-  > 

10  dek'aliters  =  1  hedoliter,        "        (^0-  ^ 

10  hec'toliters  =  1  kiloliter,         "        ("). 


344  THE  METRIC  SYSTEM. 

Note  1.  —  A  kiloliter  has  the  same  capacity  as  a  stare,  or  cu.  meter.. 
Note  2.  —  A  hectoliter  equals  about  2^  bushels. 

Numbers  denoting  capacity  are  written  in  the  same  manner 
as  numbers  denoting  length.     (Art.  545.) 

Examples. 

1.  Express  the  following  numbers  as  liters  and  add  them: 
458  centiliters,  82  dekaliters,  765  milliliters.        Ans.  825.345^ 

2.  From  a  vessel  containing  1  hectoliter  of  oil  were  drawn 
25  liters,  6  centiliters  ;  how  many  liters  remained?    Ans.  74.94'. 

3.  How  many  liters  of  wheat  can  be  put  into  a  bin  that  is  2 
meters  long,  1.3  meters  wide,  and  1.5  meters  high  ?     Ans.  3900\ 

4.  What  must  be  the  length  of  a  bin  that  is  If.  meters  wide 
and  1  meter  high,  to  contain  4000  liters  of  grain?      A71S,  2.5"\ 

MEASURES  OF  WEIGHT. 

0^4.    The  Gram  is  the  primary  unit  of  weight. 

The  Gram  is  the  weight,  in  a  vacuum,  of  a  cubic  centimeter 
of  distilled  water  at  the  temperature  when  it  is  most  dense, 
which  is  at  39-^°  Fahrenheit. 

Note  1.  — The  gram  equals  15.432  grains. 

Note  2.  —  The  new  five-cent  piece  (of  1866)  weighs  5  grams. 

Table. 

10  milligrams  (™^)  =  1  centigram,  marked  (««°«g). 

10  cen'tigrams  =  1  decigram,  "  (•^^"s). 

10  dec'igrams  =  1  gram,  "  (s). 

10  GRAMS  =  1  dekagram,  "  (^^^'^s). 

10  dek'agrams  =  1  hectogram,  "  (^°). 

10  hec'tograms  =  1  kilogram,  "  (^^)  or  (^). 

10  kil'ograms  =  1  myriagram,  "  ('"yriag). 

10  myr'iagrams  =  1  quintal,  "  (^). 

10  quintals  •    =  1  tonneau,  "  C). 

The  kilogram,  sometimes  called  the  kilo,  is  considered  the 
unit  in  weighing  gross,  heavy  articles.  The  kilogram  equals 
about  2-i  pounds  avoirdupois,  or,  more  nearly,  2.2046  pounds. 
The  tonneau  equals  a  little  more  than  2204  pounds. 


MEASURES  OF  WEIGHT. 


345 


Numbers  denoting  weight  are  written  in  the  same  manner  as 
numbers  denoting  length.     (See  Art.  545.) 

Examples. 

1.  Express  the  following  numbers  as  grams  and  add  them: 
8.5  dekagrams,  1000  centigrams,  225  decigrams.     Ans.  117.5^. 

2.  Express  the  following  numbers  as  kilograms  and  add  them  : 
7.2  hectograms,  8294  grams,  4  quintals.  Ans.  409.014''*^. 

3.  How  many  papers,  each  containing  ^  a  kilogram,  may  be 
filled  from  32  myriagrams  of  coffee?  Ans.  640  papers. 

4.  Bought  1  tonneau  of  coal  for  $12 ;  what  is  the  cost  of  1 
kilogram  of  coal  at  the  same  rate?  Ans.  12  mills. 

5.  What  weight  of  mercury  will  a  vessel  contain  whose 
capacity  is  10  cubic  centimeters,  mercury  being  13.5  times  as 
heavy  as  water.  Ans.  135  grams. 

6.  In  77.2  grams  of  gold  how  many  cubic  centimeters,  gold 
being  19.3  times  as  heavy  as  water?  Ans.  4*^"^^'^*^. 

METRIC  MEASURES    LEGALIZED.  BY  THE  UNITED  STATES 
WITH  THEIR  EQUIVALENTS  NOW  IN  USE. 

Note.  —  Although  the  equivalents  here  given  are  not  entirely  accurate, 
they  are  those  which  are  established  by  Congress  for  use  in  legal  proceed- 
ings, and  in  the  interpretation  of  contracts,  and  are  sufficiently  exact  for 
all  practical  purposes. 


Measures  of  Length. 

METRIC  DENOMINATIONS  AND  VALUES. 

EQUIVALENTS  IN  DENOMINATIONS 
IN  USE. 

Myriameter  . 
Kilometer  .    . 
Hectometer  . 
Dekameter    . 
Meter .... 
Decimeter     . 
Centimeter    . 
Millimeter     . 

10,000  m.   . 

1,000  ra.   . 

100  m.   . 

10  m.   . 

Im.   . 

.1  m.   . 

.01m.   . 

.001m.   . 

6.2137  miles. 

0.62137  mile,  or  8280  ft.  10  in. 

328  ft.  1  in. 

393.7  in. 

39.37  in. 

3.937  in. 

0.3937  in. 

0.0394  in. 

346 


THE  METRIC  SYSTEM. 


Measures  of  Surface. 


METRIC  DENOMINATIONS  AND  VALUES. 

EQUIVALENTS  IN  DENOMI- 
NATIONS IN  USE. 

Hectare   . 

A.TG          ••          ....... 

10,000  sq.  m. 

100  sq.  m. 

1  sq.  m. 

2.471  acres. 
119.6  sq.  yards. 
1550.  sq.  inches. 

Centare               •       •   •        • 

Measures  of  Capacity. 


METRIC  DENOMINATIONS  AND  VALUES. 

EQUIVALENTS  IN  DENOMINATIONS 
IN  USE. 

Names. 

No.  of 
Liters. 

Cubic 
Measure. 

Dry  Measure. 

Liquid  or  Wine 
Measure. 

Kiloliter  or  Stere 

1000 

1  cu.  m. 

1.308  cu.  yds. 

264.17  gal. 

Hectoliter    .   .   . 

100 

.1  cu.  m. 

2  bu.  3.35  pks. 

26.417  gal. 

Dekaliter.    .   .    . 

10 

10  cu.  dm. 

9.08  qts.   .    .    . 

2.6417  gal. 

Liter 

1 

1  cu.  dm. 

0.908  qt.   .    .    . 

1.0567  qts. 

Deciliter  .... 

.1 

.1  cu.  dm. 

6.1022  cu.  in. 

0.845  gUl. 

Centiliter .... 

.01 

10  cu.  cm. 

0.6102  cu.  in. 

0.338  fid  oz. 

Milliliter  .... 

.001 

.1  cu.  cm. 

0.061  cu.  in.    . 

0.27  fld  dr. 

Weights. 


METRIC  DENOMINATIONS  AND  VALUES. 

EQUIVALENTS  IN 

DENOMINATIONS 

IN  USE. 

Names. 

No.  of 
Grams. 

Weight  of  what  quan- 
tity of  water  at  max- 
imum density. 

Avoirdupois 
Weight. 

Millier  or  Tonneau    . 

Quintal 

Myriagram 

Kilogram  or  Kilo    .    . 

Hectogram 

Dekagram 

Gram 

Decigram 

Centigram 

Milligram 

1,000,000 

100,000 

10,000 

1,000 

100 

10 

1 
.1 

.01 
.001 

1  cu.  meter     . 
1  hectoliter     . 
10  liters  .   .   . 
1  liter  .... 
1  deciliter   .    . 
10  cu.  centim. 
1  cu.  centim.  . 
.1  cu.  centim. 
10  cu.  millim. 
1  cu.  millim.  . 

2204.6  pounds. 
220.46       «' 
22.046       « 
2.2046       " 
3.5274  ounces. 
0.3527       " 
15.432  grains. 
1.5432      " 
0.1543      " 
0.0154      " 

REDUCTION  OF  NUMBERS.  347 


REDUCTION   OF   NUMBERS    IN   THE    METRIC   SYSTEM   TO 
EQUIVALENTS  NOW  IN  USE. 

55^,   III.  Ex.    In  5  meters  how  many  inches?   how  many 

^^^^ '  Explanation.     Since  1  meter 

equals  39.37  inches,  5  meters  must 

opekation.  equal  5  times  39.37  inches,  and 

39.37  X5        i/.irtc/v      A  since  12  inches  equall  foot,  there 

=  16.40 rV  ft.,  Ans.  ,  ,  c   4.     ^u 

12  1^       »  must  be  as  many  feet  as  there  are 

times  12  m  5  times  39.37,  which 

is  16.403^,^  times. 

Am.  16.40y\ft. 

Examples. 

1.  In  1  meter,  2  decimeters,  how  many  feet?     Ans.  3.937  ft. 

2.  In  25  millimeters  how  many  inches  ?  Ans.  .985  in. 

3.  How  many  inches  long  is  a  silkworm  that  measures  5.2 
centimeters  in  length  ?  •  Ans.  2.047  in.4-. 

4.  What  is  the  height  of  a  person  in  feet  and  inches  whose 
height  is  1  meter,  728  millimeters?  Ans.  5  ft.  8  in.+. 

5.  In  21  ares  how  many  square  yards?    Ans.  2511.6  sq.  yds. 

6.  In  a  field  of  7  hectares,  2  ares,  how  many  acres? 

Ans.  17.346  acres-}-. 

7.  In  32  centares  how  many  square  yards  ?• 

Ans.  38.272  sq.  yds. 

8.  In  12  steres  of  wood  how  many  cubic  yards?  how  many  • 
cords?  Ans.  15.696  yds. ;  3.31  cords -f-. 

9.  In  24  kiloliters  how  many  gallons  ?         Ans,  6340.08  gal. 

10.  How  many  gallons  of  vinegar  may  be  put  into  a  cask 
containing  8  dekaliters,  2  liters?  Ans.  21.66  gals.-[-. 

11.  In  2  hectoliters  how  many  bushels  and  pecks? 

Ans.  5bu.  2.7  pks. 

12.  In  23  kilograms  how  many  pounds  avoirdupois? 

I  ;  Ans.  50.7  lbs.-)-. 

13.  In  ^8-kilograms,  7  hectograms,  how  many  pounds  avoir- 
dupois? Ans.  41.22  lbs.-|-. 

14.  In  27  tonneaux  how  many  tons?        Ans.  29.762  tons -4-. 


348 


THE  METRIC  SYSTEM. 


5Sy7»    The  following  are  some  of  the  measures  in  common 
use,  with  their  equivalents  in  measures  of  the  metric  system  :  — 


An  inch 

=  2.54  centimeters. 

A  cu.  yard 

=  .7646  cu.  meter. 

Afoot 

=  .3048  meter. 

A  cord 

=  8.624  steres. 

A  yard 

=  .1)144  meter. 

A  liquid  quart 

=  .9464  liter. 

A  rod 

=  5.029  meters. 

A  gallon 

=  3.786  liters. 

A  mile 

=  1.6093  kilometers. 

A  dry  quart 

=-1.101  liters. 

A  sq. inch 

=  6.452  sq.  centimet's. 

A  peck 

=  8.811  liters. 

A  sq.  foot 

=  .0929  sq.  meter. 

A  bushel 

=  35.24  liters. 

A  sq.  yard 

=  .8361  sq.  meter. 

An  ounce  av. 

=  28.35  grams. 

A  sq.  rod 

=  25.29  sq.  meters. 

A  pound  av. 

=  .4536  kilogram. 

An  acre 

=  .4047  hectare. 

A  ton 

r=  .9072  tonneau. 

A  sq.  mile 

==  259  hectares. 

A  grain  Troy 

=  .0648  gram. 

A  cu.  inch 

=  16.39  cu.  centimet's. 

An  ounce  Troy 

=  31.104  grams. 

A  cu.  foot 

=  .02832  cu.  meter. 

A  pound  Troy 

=  .3732  kilogram. 

KEDUCTION  OF    MEASURES   NOW   IN    USE    TO    EQUIVALENTS 
IN  THE   METRIC   SYSTEM. 

558.    III.  Ex,    In  5  feet  how  many  meters  ? 

ExPLAXATiON.     Since  1  foot  equals 
.3048  TTifttRrs.  .5  fpftf. 

Ans. 


Opekation. 

.3048  X  5  =  1.524 


.3048  meters,  5  feet  must  equal  5  times 
.3048  meters,  which  is  1.524  meters. 
Ans.  1.524". 
Examples. 


4-.  In  2  rods  how  many  meters  ? 
■%.  In  25  sq.  yards  how  many  sq.  meters  ? 
-3.  In  23  acres  how  many  hectares  ? 
~4.  In  8.2  cords  how  many  steres? 
-h.  In  9.2  liquid  quarts  how  many  liters? 
-^.  In  28  grains  how  many  grams  ? 
7.  In  3  yards  1  foot  how  many  meters  ? 
-8.  In  2  bushels  3  pecks  how  many  liters? 
9.  In  8  pounds  7  ounces  how  many  kilos? 

"^^  For  a  fuller   treatment  of  this  subject  see  Walton's  pamphlet 
edition  of  *•  The  Metric  System  of  Weights  and  Measures." 


Ans.  10.058"^. 

Ans.  20.9«<i"^. 

Ans.  9.308^*+. 

Ans.  29.725«4-. 

Ans.  8.706^+. 

Ans.   1.814^+. 

Ans.  3.048™. 

Ans.  96.921^. 

Ans.  3;427^+. 


